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▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s

▼❛st❡r ♦❢ ❙❝✐❡♥❝❡ ✐♥ ❊❧❡❝tr✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ❊r✐✈❡❧t♦♥ ●❡r❛❧❞♦ ◆❡♣♦♠✉❝❡♥♦

❉❡♣❛rt♠❡♥t ♦❢ ❊❧❡❝tr✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ❋❡❞❡r❛❧ ❯♥✐✈❡rs✐t② ♦❢ ❙ã♦ ❏♦ã♦ ❞❡❧✲❘❡✐

❆✉❣✉st ✷✵✶✺

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✶ ✴ ✽✶

❚❡❛❝❤✐♥❣ P❧❛♥

❈♦♥t❡♥t

✶ ❚❤❡ ❘❡❛❧ ❛♥❞ ❈♦♠♣❧❡① ◆✉♠❜❡rs ❙②st❡♠s ✷ ❇❛s✐❝ ❚♦♣♦❧♦❣② ✸ ◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ✹ ❈♦♥t✐♥✉✐t② ✺ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✻ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ♦❢ ❋✉♥❝t✐♦♥s ✼ ■❊❊❊ ❙t❛♥❞❛r❞ ❢♦r ❋❧♦❛t✐♥❣✲P♦✐♥t ❆r✐t❤♠❡t✐❝ ✽ ■♥t❡r✈❛❧ ❆♥❛❧②s✐s Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✷ ✴ ✽✶

❘❡❢❡r❡♥❝❡s ❘✉❞✐♥✱ ❲✳ ✭✶✾✼✻✮✱ Pr✐♥❝✐♣❧❡s ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ❛♥❛❧②s✐s✱ ▼❝●r❛✇✲❍✐❧❧ ◆❡✇ ❨♦r❦✳ ▲✐♠❛✱ ❊✳ ▲✳ ✭✷✵✶✹✮✳ ❆♥á❧✐s❡ ❘❡❛❧ ✲ ❱♦❧✉♠❡ ✶ ✲ ❋✉♥çõ❡s ❞❡ ❯♠❛ ❱❛r✐á✈❡❧✳ ✶✷ ❡❞✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ■▼P❆✳ ❖✈❡rt♦♥✱ ▼✳ ▲✳ ✭✷✵✵✶✮✱ ◆✉♠❡r✐❝❛❧ ❈♦♠♣✉t✐♥❣ ✇✐t❤ ■❊❊❊ ✢♦❛t✐♥❣ ♣♦✐♥t ❛r✐t❤♠❡t✐❝✱ ❙■❆▼✳ ■♥st✐t✉t❡ ♦❢ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❊❧❡❝tr♦♥✐❝ ❊♥❣✐♥❡❡r✐♥❣ ✭✷✵✵✽✮✱ ✼✺✹✲✷✵✵✽ ✕ ■❊❊❊ st❛♥❞❛r❞ ❢♦r ✢♦❛t✐♥❣✲♣♦✐♥t ❛r✐t❤♠❡t✐❝✳

• ♦❧❞❜❡r❣✱ ❉✳ ✭✶✾✾✶✮✱ ❲❤❛t ❊✈❡r② ❈♦♠♣✉t❡r ❙❝✐❡♥t✐st ❙❤♦✉❧❞ ❑♥♦✇

❆❜♦✉t ❋❧♦❛t✐♥❣✲♣♦✐♥t ❆r✐t❤♠❡t✐❝✱ ❈♦♠♣✉t✐♥❣ ❙✉r✈❡②s ✷✸✭✶✮✱ ✺✕✹✽✳ ▼♦♦r❡✱ ❘✳ ❊✳ ✭✶✾✼✾✮✱ ▼❡t❤♦❞s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ■♥t❡r✈❛❧ ❆♥❛❧②s✐s✱ P❤✐❧❛❞❡❧♣❤✐❛✿ ❙■❆▼✳ ◆❡♣♦♠✉❝❡♥♦✱ ❊✳ ● ✭✷✵✶✹✮✳ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ r❡❝✉rs✐✈❡ ❢✉♥❝t✐♦♥s ♦♥ ❝♦♠♣✉t❡rs✳ ❚❤❡ ❏♦✉r♥❛❧ ♦❢ ❊♥❣✐♥❡❡r✐♥❣ q✱ ■♥st✐t✉t✐♦♥ ♦❢ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❚❡❝❤♥♦❧♦❣②✱ ✶✲✸✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✸ ✴ ✽✶

❆ss❡ss♠❡♥t ■t❡♠ ❱❛❧✉❡ ❉❛t❡ ❖❜s❡r✈❛t✐♦♥ N1 ✲ ❊①❛♠ ✶ ✶✵✵ ✵✷✴✵✾✴✷✵✶✺ ❈❤❛♣t❡rs ✶ ❛♥❞ ✷ N2 ✲ ❊①❛♠ ✷ ✶✵✵ ✶✹✴✶✵✴✷✵✶✺ ❈❤❛♣t❡rs ✸ ❛♥❞ ✹✳ N3 ✲ ❊①❛♠ ✸ ✶✵✵ ✵✹✴✶✶✴✷✵✶✺ ❈❤❛♣t❡rs ✺ ❛♥❞ ✻✳ N4 ✲ ❊①❛♠ ✹ ✶✵✵ ✵✷✴✶✷✴✷✵✶✺ ❈❤❛♣t❡rs ✼ ❛♥❞ ✽✳ Ns ✲ ❙❡♠✐♥❛r ✶✵✵ ✵✾✴✶✷✴✷✵✶✺ P❛♣❡r ✰ Pr❡s❡♥t❛t✐♦♥✳ Ne ✲ ❊s♣❡❝✐❛❧ ✶✵✵ ✶✻✴✶✷✴✷✵✶✺ ❊s♣❡❝✐❛❧ ❊①❛♠

❚❛❜❧❡ ✶✿ ❆ss❡s♠❡♥t ❙❝❤❡❞✉❧❡

❙❝♦r❡✿ S = 2(N1 + N2 + N3 + N4 + Ns) 500 ❲✐t❤ Ne t❤❡ ✜♥❛❧ s❝♦r❡ ✐s✿ Sf = S + Ne 2 ✱ ♦t❤❡r✇✐s❡ Sf = S. ■❢ Sf ≥ 6.0 t❤❡♥ ❙✉❝❝❡❡❞✳ ■❢ Sf < 6.0 t❤❡♥ ❋❛✐❧❡❞✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✹ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ■♥tr♦❞✉❝t✐♦♥

✶✳ ❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s

✶✳✶ ■♥tr♦❞✉❝t✐♦♥ ❆ ❞✐s❝✉ss✐♦♥ ♦❢ t❤❡ ♠❛✐♥ ❝♦♥❝❡♣ts ♦❢ ❛♥❛❧②s✐s ✭s✉❝❤ ❛s ❝♦♥✈❡r❣❡♥❝❡✱ ❝♦♥t✐♥✉✐t②✱ ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ❛♥❞ ✐♥t❡❣r❛t✐♦♥✮ ♠✉st ❜❡ ❜❛s❡❞ ♦♥ ❛♥ ❛❝❝✉r❛t❡❧② ❞❡✜♥❡❞ ♥✉♠❜❡r ❝♦♥❝❡♣t✳ ◆✉♠❜❡r✿ ❆♥ ❛r✐t❤♠❡t✐❝❛❧ ✈❛❧✉❡ ❡①♣r❡ss❡❞ ❜② ❛ ✇♦r❞✱ s②♠❜♦❧✱♦r ✜❣✉r❡✱ r❡♣r❡s❡♥t✐♥❣ ❛ ♣❛rt✐❝✉❧❛r q✉❛♥t✐t② ❛♥❞ ✉s❡❞ ✐♥ ❝♦✉♥t✐♥❣ ❛♥❞ ♠❛❦✐♥❣ ❝❛❧❝✉❧❛t✐♦♥s✳ ✭❖①❢♦r❞ ❉✐❝t✐♦♥❛r②✮✳ ▲❡t ✉s s❡❡ ✐❢ ✇❡ r❡❛❧❧② ❦♥♦✇ ✇❤❛t ❛ ♥✉♠❜❡r ✐s✳ ❚❤✐♥❦ ❛❜♦✉t t❤✐s q✉❡st✐♦♥✿✶ Is 0.999 . . . = 1? ✭✶✮

✶❘✐❝❤♠❛♥✱ ❋✳ ✭✶✾✾✾✮ ■s ✵✳✾✾✾ ✳✳✳ ❂ ✶❄ ▼❛t❤❡♠❛t✐❝s ▼❛❣❛③✐♥❡✳ ✼✷✭✺✮✱ ✸✽✻✕✹✵✵✳ Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✺ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ■♥tr♦❞✉❝t✐♦♥

❚❤❡ s❡t N ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs ✐s ❞❡✜♥❡❞ ❜② t❤❡ P❡❛♥♦ ❆①✐♦♠s✿

✶ ❚❤❡r❡ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ s : N → N. ❚❤❡ ✐♠❛❣❡ s(n) ♦❢ ❡❛❝❤

♥❛t✉r❛❧ ♥✉♠❜❡r n ∈ N ✐s ❝❛❧❧❡❞ s✉❝❝❡ss♦r ♦❢ n.

✷ ❚❤❡r❡ ✐s ❛♥ ✉♥✐q✉❡ ♥❛t✉r❛❧ ♥✉♠❜❡r 1 ∈ N s✉❝❤ t❤❛t 1 = s(n) ❢♦r ❛❧❧

n ∈ N.

✸ ■❢ ❛ s✉❜s❡t X ⊂ N ✐s s✉❝❤ t❤❛t 1 ∈ X ❛♥❞ s(X) ⊂ X ✭t❤❛t ✐s✱

n ∈ X ⇒ s(n) ∈ X) t❤❡♥ X = N.

❚❤❡ s❡t Z = {. . . , −2, −1, 0, 1, 2 . . .} ♦❢ ✐♥t❡❣❡rs ✐s ❛ ❜✐❥❡❝t✐♦♥ f : N → Z s✉❝❤ t❤❛t f(n) = (n − 1)/2 ✇❤❡♥ n ✐s ♦❞❞ ❛♥❞ f(n) − n/2 ✇❤❡♥ n ✐s ❡✈❡♥✳ ❚❤❡ s❡t Q = {m/n; m, n ∈ Z, n = 0} ♦❢ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ♠❛② ❜❡ ✇r✐tt❡♥ ❛s f : Z × Z∗ → Q s✉❝❤ t❤❛t Z∗ = Z − {0} ❛♥❞ f(m, n) = m/n✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✻ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ■♥tr♦❞✉❝t✐♦♥

❚❤❡ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ❛r❡ ✐♥❛❞❡q✉❛t❡ ❢♦r ♠❛♥② ♣✉r♣♦s❡s✱ ❜♦t❤ ❛s ❛ ✜❡❧❞ ❛♥❞ ❛s ❛♥ ♦r❞❡r❡❞ s❡t✳ ❋♦r ✐♥st❛♥❝❡✱ t❤❡r❡ ✐s ♥♦ r❛t✐♦♥❛❧ p s✉❝❤ t❤❛t p2 = 2✳ ❆♥ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡r ✐s ✇r✐tt❡♥ ❛s ✐♥✜♥✐t❡ ❞❡❝✐♠❛❧ ❡①♣❛♥s✐♦♥✳ ❚❤❡ s❡q✉❡♥❝❡ ✶✱ ✶✳✹✱ ✶✳✹✶✱ ✶✳✹✶✹✱ ✶✳✹✶✹✷ . . . t❡♥❞s t♦ √ 2✳ ❲❤❛t ✐s ✐t t❤❛t t❤✐s s❡q✉❡♥❝❡ t❡♥❞s t♦❄ ❲❤❛t ✐s ❛♥ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡r❄ ❚❤✐s s♦rt ♦❢ q✉❡st✐♦♥ ❝❛♥ ❜❡ ❛♥s✇❡r❡❞ ❛s s♦♦♥ ❛s t❤❡ s♦✲❝❛❧❧❡❞ ✏r❡❛❧ ♥✉♠❜❡r s②st❡♠✑ ✐s ❝♦♥str✉❝t❡❞✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✼ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ■♥tr♦❞✉❝t✐♦♥

❊①❛♠♣❧❡ ✶

❲❡ ♥♦✇ s❤♦✇ t❤❛t t❤❡ ❡q✉❛t✐♦♥ p2 = 2 ✭✷✮ ✐s ♥♦t s❛t✐s✜❡❞ ❜② ❛♥② r❛t✐♦♥❛❧ p✳ ■❢ t❤❡r❡ ✇❡r❡ s✉❝❤ ❛ p✱ ✇❡ ❝♦✉❧❞ ✇r✐t❡ p = m/n ✇❤❡r❡ m ❛♥❞ n ❛r❡ ✐♥t❡❣❡rs t❤❛t ❛r❡ ♥♦t ❜♦t❤ ❡✈❡♥✳ ▲❡t ✉s ❛ss✉♠❡ t❤✐s ✐s ❞♦♥❡✳ ❚❤❡♥ ✭✷✮ ✐♠♣❧✐❡s m2 = 2n2. ✭✸✮ ❚❤✐s s❤♦✇s t❤❛t m2 ✐s ❡✈❡♥✳ ❍❡♥❝❡ m ✐s ❡✈❡♥ ✭✐❢ m ✇❡r❡ ♦❞❞✱ m2 ✇♦✉❧❞ ❜❡ ♦❞❞✮✱ ❛♥❞ s♦ m2 ✐s ❞✐✈✐s✐❜❧❡ ❜② ✹✳ ■t ❢♦❧❧♦✇s t❤❛t t❤❡ r✐❣❤t s✐❞❡ ♦❢ ✭✸✮ ✐s ❞✐✈✐s✐❜❧❡ ❜② ✹✱ s♦ t❤❛t n2 ✐s ❡✈❡♥✱ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t n ✐s ❡✈❡♥✳ ❚❤✉s t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t ✭✷✮ ❤♦❧❞s t❤✉s ❧❡❛❞s t♦ t❤❡ ❝♦♥❝❧✉s✐♦♥ t❤❛t ❜♦t❤ m ❛♥❞ n ❛r❡ ❡✈❡♥✱ ❝♦♥tr❛r② t♦ ♦✉r ❝❤♦✐❝❡ ♦❢ m ❛♥❞ n✳ ❍❡♥❝❡ ✭✷✮ ✐s ✐♠♣♦ss✐❜❧❡ ❢♦r r❛t✐♦♥❛❧ p✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✽ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ■♥tr♦❞✉❝t✐♦♥

▲❡t ✉s ❡①❛♠✐♥❡ ♠♦r❡ ❝❧♦s❡❧② t❤❡ ❊①❛♠♣❧❡ ✶✳ ▲❡t A ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ♣♦s✐t✐✈❡ r❛t✐♦♥❛❧s p s✉❝❤ t❤❛t p2 < 2 ❛♥❞ ❧❡t B ❝♦♥s✐st ♦❢ ❛❧❧ ♣♦s✐t✐✈❡ r❛t✐♦♥❛❧s p s✉❝❤ t❤❛t p2 > 2. ❲❡ s❤❛❧❧ s❤♦✇ t❤❛t A ❝♦♥t❛✐♥s ♥♦ ❧❛r❣❡st ♥✉♠❜❡r ❛♥❞ B ❝♦♥t❛✐♥s ♥♦ s♠❛❧❧❡st✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❡✈❡r② p ∈ A ✇❡ ❝❛♥ ✜♥❞ ❛ r❛t✐♦♥❛❧ q ∈ A s✉❝❤ t❤❛t p < q, ❛♥❞ ❢♦r ❡✈❡r② p ∈ B ✇❡ ❝❛♥ ✜♥❞ ❛ r❛t✐♦♥❛❧ q ∈ B s✉❝❤ t❤❛t q < p. ▲❡t ❡❛❝❤ r❛t✐♦♥❛❧ p > 0 ❜❡ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ♥✉♠❜❡r q = p − p2 − 2 p + 2 = 2p + 2 p + 2 . ✭✹✮ ❛♥❞ q2 = (2p + 2)2 (p + 2)2 . ✭✺✮

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✾ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ■♥tr♦❞✉❝t✐♦♥

▲❡t ✉s r❡✇r✐t❡ q = p − p2 − 2 p + 2 ✭✻✮ ▲❡t ✉s s✉❜tr❛❝t ✷ ❢r♦♠ ❜♦t❤ s✐❞❡s ♦❢ ✭✻✮ q2 − 2 = (2p + 2)2 (p + 2)2 − 2(p + 2)2 (p + 2)2 q2 − 2 = (4p2 + 8p + 8) − (2p2 + 8p + 4) (p + 2)2 q2 − 2 = 2(p2 + 2) (p + 2)2 . ✭✼✮ ■❢ p ∈ A t❤❡♥ p2 − 2 < 0, ✭✻✮ s❤♦✇s t❤❛t q > p✱ ❛♥❞ ✭✼✮ s❤♦✇s t❤❛t q2 < 2. ❚❤✉s q ∈ A. ■❢ p ∈ B t❤❡♥ p2 − 2 > 0, ✭✻✮ s❤♦✇s t❤❛t 0 < q < p, ❛♥❞ ✭✼✮ s❤♦✇s t❤❛t q2 > 2. ❚❤✉s q ∈ B.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✶✵ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ■♥tr♦❞✉❝t✐♦♥

■♥ t❤✐s s❧✐❞❡ ✇❡ s❤♦✇ t✇♦ ✇❛②s t♦ ❛♣♣r♦❛❝❤ √ 2✳ ◆❡✇t♦♥✬s ♠❡t❤♦❞ √ 2 = lim

n→∞ xn+1 = xn

2 + 1 xn ✭✽✮ ✇❤✐❝❤ ♣r♦❞✉❝❡s t❤❡ s❡q✉❡♥❝❡ ❢♦r x0 = 1

❚❛❜❧❡ ✷✿ ❙❡q✉❡♥❝❡ ♦❢ xn ♦❢ ✭✽✮

n xn ✭❢r❛❝t✐♦♥✮ xn ✭❞❡❝✐♠❛❧✮ ✵ ✶ ✶ ✶ 3 2 ✶✳✺ ✷ 17 12 1.41¯ 6 ✸ 577 408 1.4142 . . .

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✶✶ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ■♥tr♦❞✉❝t✐♦♥

◆♦✇ ❧❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❣✐✈❡♥ ❜② √ 2 = 1 + 1 1 + 2 1 + 2 1 + . . . ✭✾✮ ✇❤✐❝❤ ♣r♦❞✉❝❡s t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡

❚❛❜❧❡ ✸✿ ❙❡q✉❡♥❝❡ ♦❢ xn ♦❢ ✭✾✮

n xn ✭❢r❛❝t✐♦♥✮ xn ✭❞❡❝✐♠❛❧✮ ✵ ✶ ✶ ✶ 3 2 ✶✳✺ ✷ 7 5 1.4 ✸ 17 12 1.41¯ 6 . . .

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✶✷ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ■♥tr♦❞✉❝t✐♦♥

❘❡♠❛r❦ ✶

❚❤❡ r❛t✐♦♥❛❧ ♥✉♠❜❡r s②st❡♠ ❤❛s ❝❡rt❛✐♥ ❣❛♣s✱ ✐♥ s♣✐t❡ t❤❡ ❢❛❝t t❤❛t ❜❡t✇❡❡♥ ❛♥② t✇♦ r❛t✐♦♥❛❧ t❤❡r❡ ✐s ❛♥♦t❤❡r✿ ✐❢ r < s t❤❡♥ r < (r + s)/2 < s✳ ❚❤❡ r❡❛❧ ♥✉♠❜❡r s②st❡♠ ✜❧❧ t❤❡s❡ ❣❛♣s✳

❉❡✜♥✐t✐♦♥ ✶

■❢ A ✐s ❛♥② s❡t✱ ✇❡ ✇r✐t❡ x ∈ A t♦ ✐♥❞✐❝❛t❡ t❤❛t x ✐s ❛ ♠❡♠❜❡r ♦❢ A✳ ■❢ x ✐s ♥♦t ❛ ♠❡♠❜❡r ♦❢ A✱ ✇❡ ✇r✐t❡✿ x / ∈ A✳

❉❡✜♥✐t✐♦♥ ✷

❚❤❡ s❡t ✇❤✐❝❤ ❝♦♥t❛✐♥s ♥♦ ❡❧❡♠❡♥t ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ t❤❡ ❡♠♣t② s❡t✳ ■❢ ❛ s❡t ❤❛s ❛t ❧❡❛st ♦♥❡ ❡❧❡♠❡♥t✱ ✐t ✐s ❝❛❧❧❡❞ ♥♦♥❡♠♣t②✳

❉❡✜♥✐t✐♦♥ ✸

■❢ ❡✈❡r② ❡❧❡♠❡♥t ♦❢ A ✐s ❛♥ ❡❧❡♠❡♥t ♦❢ B✱ ✇❡ s❛② t❤❛t A ✐s ❛ s✉❜s❡t ♦❢ B✳ ❛♥❞ ✇r✐t❡ A ⊂ B✱ ♦r B ⊃ A✳ ■❢✱ ✐♥ ❛❞❞✐t✐♦♥✱ t❤❡r❡ ✐s ❛♥ ❡❧❡♠❡♥t ♦❢ B ✇❤✐❝❤ ✐s ♥♦t ✐♥ A✱ t❤❡♥ A ✐s s❛✐❞ t♦ ❜❡ ❛ ♣r♦♣❡r s✉❜s❡t ♦❢ B✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✶✸ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ❖r❞❡r❡❞ ❙❡ts

✶✳✷ ❖r❞❡r❡❞ ❙❡ts

❉❡✜♥✐t✐♦♥ ✹

▲❡t S ❜❡ ❛ s❡t✳ ❆♥ ♦r❞❡r ♦♥ S ✐s ❛ r❡❧❛t✐♦♥✱ ❞❡♥♦t❡ ❜② <✱ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ♣r♦♣❡rt✐❡s✿

✶ ■❢ x ∈ S ❛♥❞ y ∈ S t❤❡♥ ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ ♦❢ t❤❡ st❛t❡♠❡♥ts

x < y, x = y, y < x ✐s tr✉❡✳

✷ ■❢ x, y, z ∈ S✱ ✐❢ x < y ❛♥❞ y < z, t❤❡♥ x < z.

❚❤❡ ♥♦t❛t✐♦♥ x ≤ y ✐♥❞✐❝❛t❡s t❤❛t x < y ♦r x = y, ✇✐t❤♦✉t s♣❡❝✐❢②✐♥❣ ✇❤✐❝❤ ♦❢ t❤❡s❡ t✇♦ ✐s t♦ ❤♦❧❞✳

❉❡✜♥✐t✐♦♥ ✺

❆♥ ♦r❞❡r❡❞ s❡t ✐s ❛ s❡t S ✐♥ ✇❤✐❝❤ ❛♥ ♦r❞❡r ✐s ❞❡✜♥❡❞✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✶✹ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ❖r❞❡r❡❞ ❙❡ts

❉❡✜♥✐t✐♦♥ ✻

❙✉♣♣♦s❡ S ✐s ❛♥ ♦r❞❡r❡❞ s❡t✱ ❛♥❞ E ⊂ S. ■❢ t❤❡r❡ ❡①✐sts ❛ β ∈ S s✉❝❤ t❤❛t x ≤ β ❢♦r ❡✈❡r② x ∈ E✱ ✇❡ s❛② t❤❛t E ✐s ❜♦✉♥❞❡❞ ❛❜♦✈❡✱ ❛♥❞ ❝❛❧❧ β ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦❢ E✳ ▲♦✇❡r ❜♦✉♥❞ ❛r❡ ❞❡✜♥❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛② ✭✇✐t❤ ≥ ✐♥ ♣❧❛❝❡ ♦❢ ≤).

❉❡✜♥✐t✐♦♥ ✼

❙✉♣♣♦s❡ S ✐s ❛♥ ♦r❞❡r❡❞ s❡t✱ E ⊂ S, ❛♥❞ E ✐s ❜♦✉♥❞❡❞ ❛❜♦✈❡✳ ❙✉♣♣♦s❡ t❤❡r❡ ❡①✐sts ❛♥ α ∈ S ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿

✶ α ✐s ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦❢ E✳ ✷ ■❢ γ < α t❤❡♥ γ ✐s ♥♦t ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦❢ E✳

❚❤❡♥ α ✐s ❝❛❧❧❡❞ t❤❡ ❧❡❛st ✉♣♣❡r ❜♦✉♥❞ ♦❢ E ♦r t❤❡ s✉♣r❡♠✉♠ ♦❢ E✱ ❛♥❞ ✇❡ ✇r✐t❡ α = sup E.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✶✺ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ❖r❞❡r❡❞ ❙❡ts

❉❡✜♥✐t✐♦♥ ✽

❚❤❡ ❣r❡❛t❡st ❧♦✇❡r ❜♦✉♥❞✱ ♦r ✐♥✜♠✉♠✱ ♦❢ ❛ s❡t E ✇❤✐❝❤ ✐s ❜♦✉♥❞❡❞ ❜❡❧♦✇ ✐s ❞❡✜♥❡❞ ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r ♦❢ ❉❡✜♥✐t✐♦♥ ✼✿ ❚❤❡ st❛t❡♠❡♥t α = inf E. ♠❡❛♥s t❤❛t α ✐s ❛ ❧♦✇❡r ❜♦✉♥❞ ♦❢ E ❛♥❞ t❤❛t ♥♦ β ✇✐t❤ β > α ✐s ❛ ❧♦✇❡r ❜♦✉♥❞ ♦❢ E.

❊①❛♠♣❧❡ ✷

■❢ α = sup E ❡①✐sts✱ t❤❡♥ α ♠❛② ♦r ♠❛② ♥♦t ❜❡ ❛ ♠❡♠❜❡r ♦❢ E✳ ❋♦r ✐♥st❛♥❝❡✱ ❧❡t E1 ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ r ∈ Q ✇✐t❤ r < 0. ▲❡t E2 ❜❡ t❤❡ s❡t ♦❢ ♦❢ ❛❧❧ r ∈ Q ✇✐t❤ r ≤ 0✳ ❚❤❡♥ sup E1 = sup E2 = 0, ❛♥❞ 0 / ∈ E1✱ 0 in E2.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✶✻ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ❖r❞❡r❡❞ ❙❡ts

❉❡✜♥✐t✐♦♥ ✾

❆♥ ♦r❞❡r❡❞ s❡t S ✐s s❛✐❞ t♦ ❤❛✈❡ t❤❡ ❧❡❛st✲✉♣♣❡r✲❜♦✉♥❞ ♣r♦♣❡rt② ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s tr✉❡✿ ■❢ E ⊂ S, E ✐s ♥♦t ❡♠♣t②✱ ❛♥❞ E ✐s ❜♦✉♥❞❡❞ ❛❜♦✈❡✱ t❤❡♥ sup E ❡①✐sts ✐♥S.

❚❤❡♦r❡♠ ✶

❙✉♣♣♦s❡ S ✐s ❛♥ ♦r❞❡r❡❞ s❡t ✇✐t❤ t❤❡ ❧❡❛st✲✉♣♣❡r✲❜♦✉♥❞ ♣r♦♣❡rt②✱ B ⊂ S, B ✐s ♥♦t ❡♠♣t②✱ ❛♥❞ B ✐s ❜♦✉♥❞❡❞ ❜❡❧♦✇✳ ▲❡t L ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ❧♦✇❡r ❜♦✉♥❞s ♦❢ B✳ ❚❤❡♥ α = sup L ❡①✐sts ✐♥ S ❛♥❞ α = inf B.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✶✼ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ❋✐❡❧❞s

✶✳✸ ❋✐❡❧❞s

❉❡✜♥✐t✐♦♥ ✶✵

❆ ✜❡❧❞ ✐s ❛ s❡t F ✇✐t❤ t✇♦ ♦♣❡r❛t✐♦♥s✱ ❝❛❧❧❡❞ ❛❞❞✐t✐♦♥ ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✇❤✐❝❤ s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ s♦✲❝❛❧❧❡❞ ✏✜❡❧❞ ❛①✐♦♠s✑ ✭❆✮✱ ✭▼✮ ❛♥❞ ✭❉✮✿ ✭❆✮ ❆①✐♦♠s ❢♦r ❛❞❞✐t✐♦♥ ✭❆✶✮ ■❢ x ∈ F ❛♥❞ y ∈ F, t❤❡♥ t❤❡✐r s✉♠ x + y ✐s ✐♥ F. ✭❆✷✮ ❆❞❞✐t✐♦♥ ✐s ❝♦♠♠✉t❛t✐✈❡✿ x + y = y + x ❢♦r ❛❧❧ x, y ∈ F. ✭❆✸✮ ❆❞❞✐t✐♦♥ ✐s ❛ss♦❝✐❛t✐✈❡✿ (x + y) + z = x + (y + z) ❢♦r ❛❧❧ x, y, z ∈ F. ✭❆✹✮ F ❝♦♥t❛✐♥s ❛♥ ❡❧❡♠❡♥t ✵ s✉❝❤ t❤❛t 0 + x = x ❢♦r ❡✈❡r② x ∈ F. ✭❆✺✮ ❚♦ ❡✈❡r② x ∈ F ❝♦rr❡s♣♦♥❞s ❛♥ ❡❧❡♠❡♥t −x ∈ F s✉❝❤ t❤❛t x + (−x) = 0. ✭▼✮ ❆①✐♦♠s ❢♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✭▼✶✮ ■❢ x ∈ F ❛♥❞ y ∈ F, t❤❡♥ t❤❡✐r ♣r♦❞✉❝t xy ✐s ✐♥ F. ✭▼✷✮ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ❝♦♠♠✉t❛t✐✈❡✿ xy = yx ❢♦r ❛❧❧ x, y ∈ F.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✶✽ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ❋✐❡❧❞s

✭▼✸✮ ▼✉❧t✐♣❧✐❝❛t✐✈❡ ✐s ❛ss♦❝✐❛t✐✈❡✿ (xy)z = x(yz) ❢♦r ❛❧❧ x, y, z ∈ F. ✭▼✹✮ F ❝♦♥t❛✐♥s ❛♥ ❡❧❡♠❡♥t 1 = 0 s✉❝❤ t❤❛t 1x = x ❢♦r ❡✈❡r② x ∈ F. ✭▼✺✮ ■❢ x ∈ F ❛♥❞ x = 0 t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ❡❧❡♠❡♥t 1/x ∈ F s✉❝❤ t❤❛t x · (1/x) = 1. ✭❉✮ ❚❤❡ ❞✐str✐❜✉t✐✈❡ ❧❛✇ x(y + z) = xy + xz ❤♦❧❞s ❢♦r ❛❧❧ x, y, z ∈ F.

❉❡✜♥✐t✐♦♥ ✶✶

❆♥ ♦r❞❡r❡❞ ✜❡❧❞ ✐s ❛ ✜❡❧❞ F ✇❤✐❝❤ ✐s ❛❧s♦ ❛♥ ♦r❞❡r❡❞ s❡t✱ s✉❝❤ t❤❛t

✶ x + y < x + z ✐❢ x, y, z ∈ F ❛♥❞ y < z. ✷ xy > 0 ✐❢ x ∈ F, y ∈ F, x > 0, ❛♥❞ y > 0. Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✶✾ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ❚❤❡ r❡❛❧ ✜❡❧❞

✶✳✹ ❚❤❡ r❡❛❧ ✜❡❧❞

❚❤❡♦r❡♠ ✷

❚❤❡r❡ ❡①✐sts ❛♥ ♦r❞❡r❡❞ ✜❡❧❞ R ✇❤✐❝❤ ❤❛s t❤❡ ❧❡❛st✲✉♣♣❡r✲❜♦✉♥❞ ♣r♦♣❡rt②✳ ▼♦r❡♦✈❡r✱ R ❝♦♥t❛✐♥s Q ❛s ❛ s✉❜✜❡❧❞✳

• 2
• 1

1 2 3 4 1/2 π e

√2

❋✐❣✉r❡ ✶✿ ❘❡❛❧ ▲✐♥❡

❚❤❡♦r❡♠ ✸

✭❛✮ ■❢ x ∈ R✱ ❛♥❞ x > 0, t❤❡♥ t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n s✉❝❤ t❤❛t nx > y✳ ✭❜✮ ■❢ x ∈ R✱ ❛♥❞ x < y✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ p ∈ Q s✉❝❤ t❤❛t x < p < y.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✷✵ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ❚❤❡ r❡❛❧ ✜❡❧❞

❚❤❡♦r❡♠ ✹

❋♦r ❡✈❡r② r❡❛❧ x > 0 ❛♥❞ ❡✈❡r② ✐♥t❡❣❡r n > 0 t❤❡r❡ ✐s ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ r❡❛❧ y s✉❝❤ t❤❛t yn = x. Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✹✿ ❚❤❛t t❤❡r❡ ✐s ❛t ♠♦st ♦♥❡ s✉❝❤ y ✐s ❝❧❡❛r✱ s✐♥❝❡ 0 < y1 < y2, ✐♠♣❧✐❡s yn

1 < yn 2 ✳

▲❡t E ❜❡ t❤❡ s❡t ❝♦♥s✐st✐♥❣ ♦❢ ❛❧❧ ♣♦s✐t✐✈❡ r❡❛❧ ♥✉♠❜❡rs t s✉❝❤ t❤❛t tn < x. ■❢ t = x/(1 + x) t❤❡♥ 0 < t < 1✳ ❍❡♥❝❡ tn < t < x. ❚❤✉s t ∈ E✱ ❛♥❞ E ✐s ♥♦t ❡♠♣t②✳ ❚❤✉s 1 + x ✐s ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦❢ E. ■❢ t > 1 + x t❤❡♥ tn > t > x, s♦ t❤❛t t / ∈ E. ❚❤✉s 1 + x ✐s ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦❢ E ❛♥❞ t❤❡r❡ ✐s y = sup E. ❚♦ ♣r♦✈❡ t❤❛t yn = x ✇❡ ✇✐❧❧ s❤♦✇ t❤❛t ❡❛❝❤ ♦❢ t❤❡ ✐♥❡q✉❛❧✐t✐❡s yn < x ❛♥❞ yn > x ❧❡❛❞s t♦ ❝♦♥tr❛❞✐❝t✐♦♥✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✷✶ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ❚❤❡ r❡❛❧ ✜❡❧❞

❚❤❡ ✐❞❡♥t✐t② bn − an = (b − a)(bn−1 + bn−2a + · · · an−1) ②✐❡❧❞s t❤❡ ✐♥❡q✉❛❧✐t② bn − an < (b − a)nbn−1 ✇❤❡♥ 0 < a < b. ❆ss✉♠❡ yn < x. ❈❤♦♦s❡ h s♦ t❤❛t 0 < h < 1 ❛♥❞ h < x − yn n(y + 1)n−1 . P✉t a = y, b = y + h. ❚❤❡♥ (y + h)n − yn < hn(y + h)n−1 < hn(y + 1)n−1 < x − yn. ❚❤✉s (y + h)n < x, ❛♥❞ y + h ∈ E. ❙✐♥❝❡ y + h > y, t❤✐s ❝♦♥tr❛❞✐❝ts t❤❡ ❢❛❝t t❤❛t y ✐s ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦❢ E. ❆ss✉♠❡ yn > x. P✉t k = yn − x nyn−1 . ❚❤❡♥ 0 < k < y. ■❢ t ≥ y − k, ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t yn − tn ≥ yn − (y − k)n < knyn−1 = yn − x.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✷✷ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ❚❤❡ r❡❛❧ ✜❡❧❞

❚❤✉s tn > x, ❛♥❞ t / ∈ E. ■t ❢♦❧❧♦✇s t❤❛t y − k ✐s ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦❢

• E. ❇✉t y − k < y, ✇❤✐❝❤ ❝♦♥tr❛❞✐❝ts t❤❡ ❢❛❝t t❤❛t y ✐s t❤❡ ❧❡❛st

✉♣♣❡r ❜♦✉♥❞ ♦❢ E. ❍❡♥❝❡ yn = x, ❛♥❞ t❤❡ ♣r♦♦❢ ✐s ❝♦♠♣❧❡t❡✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✷✸ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ❚❤❡ r❡❛❧ ✜❡❧❞

❉❡✜♥✐t✐♦♥ ✶✷

▲❡t x > 0 ❜❡ r❡❛❧✳ ▲❡t no ❜❡ t❤❡ ❧❛r❣❡st ✐♥t❡❣❡r s✉❝❤ t❤❛t n0 ≤ x. ❍❛✈✐♥❣ ❝❤♦s❡♥ n0, n1, . . . , nk−1✱ ❧❡t nk ❜❡ t❤❡ ❧❛r❣❡st ✐♥t❡❣❡r s✉❝❤ t❤❛t n0 + n1 10 + · · · + nk 10k ≤ x. ▲❡t E ❜❡ t❤❡ s❡t ♦❢ t❤❡s❡ ♥✉♠❜❡rs n0 + n1 10 + · · · + nk 10k (k = 0, 1, 2, . . .). ✭✶✵✮ ❚❤❡♥ x = sup E. ❚❤❡ ❞❡❝✐♠❛❧ ❡①♣❛♥s✐♦♥ ♦❢ x ✐s n0 · n1n2n3 · · · . ✭✶✶✮

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✷✹ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ❚❤❡ ❡①t❡♥❞❡❞ r❡❛❧ ♥✉♠❜❡r s②st❡♠

✶✳✺ ❚❤❡ ❡①t❡♥❞❡❞ r❡❛❧ ♥✉♠❜❡r s②st❡♠

❉❡✜♥✐t✐♦♥ ✶✸

❚❤❡ ❡①t❡♥❞❡❞ r❡❛❧ ♥✉♠❜❡r s②st❡♠ ❝♦♥s✐sts ♦❢ t❤❡ r❡❛❧ ✜❡❧❞ R ❛♥❞ t✇♦ s②♠❜♦❧s✿ +∞ ❛♥❞ −∞✳ ❲❡ ♣r❡s❡r✈❡ t❤❡ ♦r✐❣✐♥❛❧ ♦r❞❡r ✐♥ R, ❛♥❞ ❞❡✜♥❡ +∞ < x < −∞ ❢♦r ❡✈❡r② x ∈ R. ❆♥ ✉s✉❛❧ s②♠❜♦❧ ❢♦r t❤❡ ❡①t❡♥❞❡❞ r❡❛❧ ♥✉♠❜❡r s②st❡♠ ✐s ¯ R✳ +∞ ✐s ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦❢ ❡✈❡r② s✉❜s❡t ♦❢ t❤❡ ❡①t❡♥❞❡❞ r❡❛❧ ♥✉♠❜❡r s②st❡♠✱ ❛♥❞ t❤❛t ❡✈❡r② ♥♦♥❡♠♣t② s✉❜s❡t ❤❛s ❛ ❧❡❛st ✉♣♣❡r ❜♦✉♥❞✳ ❚❤❡ s❛♠❡ r❡♠❛r❦s ❛♣♣❧② t♦ ❧♦✇❡r ❜♦✉♥❞s✳ ❚❤❡ ❡①t❡♥❞❡❞ r❡❛❧ ♥✉♠❜❡r s②st❡♠ ❞♦❡s ♥♦t ❢♦r♠ ❛ ✜❡❧❞✳ ■t ✐s ❝✉st♦♠❛r② t♦ ♠❛❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥✈❡♥t✐♦♥s✿ ✭❛✮ ■❢ x ✐s r❡❛❧ t❤❡♥ x + ∞ = ∞, x − ∞ = −∞, x +∞ = x −∞ = 0. ✭❜✮ ■❢ x > 0 t❤❡♥ x · (+∞) = +∞, x · (−∞) = −∞.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✷✺ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ❚❤❡ ❝♦♠♣❧❡① ✜❡❧❞

✶✳✻ ❚❤❡ ❝♦♠♣❧❡① ✜❡❧❞ ✭❝✮ ■❢ x < 0 t❤❡♥ x · (+∞) = −∞, x · (−∞) = +∞.

❉❡✜♥✐t✐♦♥ ✶✹

❆ ❝♦♠♣❧❡① ♥✉♠❜❡r ✐s ❛♥ ♦r❞❡r❡❞ ♣❛✐r (a, b) ♦❢ r❡❛❧ ♥✉♠❜❡rs✳ ▲❡t x = (a, b), y = (c, d) ❜❡ t✇♦ ❝♦♠♣❧❡① ♥✉♠❜❡rs✳ ❲❡ ❞❡✜♥❡ x + y = (a + c, b + d), xy = (ac − bd, ad + bc). i = (0, 1). i2 = −1. ■❢ a ❛♥❞ b ❛r❡ r❡❛❧✱ t❤❡♥ (a, b) = a + bi✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✷✻ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ❊✉❝❧✐❞❡❛♥ ❙♣❛❝❡

✶✳✼ ❊✉❝❧✐❞❡❛♥ ❙♣❛❝❡

❉❡✜♥✐t✐♦♥ ✶✺

❋♦r ❡❛❝❤ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r k✱ ❧❡t Rk ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ♦r❞❡r❡❞ k✲t✉♣❧❡s x = (x1, x2, . . . , xk), ✇❤❡r❡ x1, . . . , xk ❛r❡ r❡❛❧ ♥✉♠❜❡rs ❝❛❧❧❡❞ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ x✳ ❆❞❞✐t✐♦♥ ♦❢ ✈❡❝t♦rs✿ x + y = (x1 + y1, . . . , xk + yk)✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❛ ✈❡❝t♦r ❜② ❛ r❡❛❧ ♥✉♠❜❡r ✭s❝❛❧❛r✮✿ αx = (αx1, . . . , αxk). ■♥♥❡r ♣r♦❞✉❝t✿ x · y = k

i=1 xiyi.

◆♦r♠✿ |x| = (x · x)1/2 = k

1 x2 i

1/2 ✳ ❚❤❡ str✉❝t✉r❡ ♥♦✇ ❞❡✜♥❡❞ ✭t❤❡ ✈❡❝t♦r s♣❛❝❡ Rk ✇✐t❤ t❤❡ ❛❜♦✈❡ ♣r♦❞✉❝t ❛♥❞ ♥♦r♠✮ ✐s ❝❛❧❧❡❞ ❊✉❝❧✐❞❡❛♥ k✲s♣❛❝❡✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✷✼ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ❊✉❝❧✐❞❡❛♥ ❙♣❛❝❡

❚❤❡♦r❡♠ ✺

❙✉♣♣♦s❡ x, y, z ∈ Rk ❛♥❞ α ✐s r❡❛❧✳ ❚❤❡♥

✶ |x| ≥ 0; ✷ |x| = 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢ |x = 0|; ✸ |αx| = |α||x|; ✹ |x · y| ≤ |x||y|❀ ✺ |x + y| ≤ |x| + |y|; ✻ |x − z| ≤ |x − y| + |x − z|.

■t❡♠s ✶✱✷ ❛♥❞ ✻ ♦❢ ❚❤❡♦r❡♠ ✺ ✇✐❧❧ ❛❧❧♦✇ ✉s t♦ r❡❣❛r❞ Rk ❛s ❛ ♠❡tr✐❝ s♣❛❝❡✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✷✽ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ❊✉❝❧✐❞❡❛♥ ❙♣❛❝❡

❊①❡r❝✐s❡s ❈❤❛♣t❡r ✶

✭✶✮ ▲❡t t❤❡ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs 1/n ✇❤❡r❡ n ∈ N✳ ❉♦❡s t❤✐s s❡q✉❡♥❝❡ ❤❛✈❡ ❛♥ ✐♥✜♠✉♠❄ ■❢ ✐t ❤❛s✱ ✇❤❛t ✐s ✐t❄ ❊①♣❧❛✐♥ ②♦✉r r❡s✉❧t ❛♥❞ s❤♦✇ ✐❢ ✐t ✐s ♥❡❝❡ss❛r② ❛♥② ♦t❤❡r ❝♦♥❞✐t✐♦♥✳ ✭✷✮ ❈♦♠♠❡♥t t❤❡ ❛ss✉♠♣t✐♦♥✿ ❊✈❡r② ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡r ✐s t❤❡ ❧✐♠✐t ♦❢ ♠♦♥♦t♦♥✐❝ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ✭❋❡rr❛r✱ ✶✾✸✽✱ ♣✳✷✵✮✳ ✭✸✮ Pr♦✈❡ ❚❤❡♦r❡♠ ✶✳ ✭✹✮ Pr♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❛✮ ■❢ x + y = x + z t❤❡♥ y = z✳ ❜✮ ■❢ x + y = x t❤❡♥ y = 0. ❝✮ ■❢ x + y = 0 t❤❡♥ y = −x. ❞✮ −(−x) = x✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✷✾ ✴ ✽✶

❚❤❡ r❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ♥✉♠❜❡r s②st❡♠s ❊✉❝❧✐❞❡❛♥ ❙♣❛❝❡

✭✺✮ Pr♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❛✮ ■❢ x > 0 t❤❡♥ −x < 0, ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ❜✮ ■❢ x > 0 ❛♥❞ y < z t❤❡♥ xy < xz✳ ❝✮ ■❢ x < 0 ❛♥❞ y < z t❤❡♥ xy > xz. ❞✮ ■❢ x = 0 t❤❡♥ x2 > 0. ❡✮ ■❢ 0 < x < y t❤❡♥ 0 < 1/y < 1/x. ✭✻✮ Pr♦✈❡ t❤❡ ❚❤❡♦r❡♠ ✷✳ ✭❖♣t✐♦♥❛❧✮ ✭✼✮ Pr♦✈❡ t❤❡ ❚❤❡♦r❡♠ ✸✳ ✭✽✮ ❲r✐t❡ ❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐str✐❜✉t✐♦♥ ❧❛✇ ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r ♦❢ ❉❡✜♥✐t✐♦♥ ✶✽ ❢♦r t❤❡ ❝♦♠♣❧❡① ✜❡❧❞✳ ✭✾✮ ❲❤❛t ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ R ❛♥❞ ¯ R? ✭✶✵✮ Pr♦✈❡ t❤❡ r❡✈❡rs❡ tr✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t②✿ ||a| − |b|| ≤ |a − b|.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✸✵ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② ❋✐♥✐t❡✱ ❈♦✉♥t❛❜❧❡✱ ❛♥❞ ❯♥❝♦✉♥t❛❜❧❡ ❙❡ts

✷✳ ❇❛s✐❝ ❚♦♣♦❧♦❣②

✷✳✶ ❋✐♥✐t❡✱ ❈♦✉♥t❛❜❧❡✱ ❛♥❞ ❯♥❝♦✉♥t❛❜❧❡ ❙❡ts

❉❡✜♥✐t✐♦♥ ✶✻

❈♦♥s✐❞❡r t✇♦ s❡ts A ❛♥❞ B ✱ ✇❤♦s❡ ❡❧❡♠❡♥ts ♠❛② ❜❡ ❛♥② ♦❜❥❡❝ts ✇❤❛ts♦❡✈❡r✱ ❛♥❞ s✉♣♣♦s❡ t❤❛t ✇✐t❤ ❡❛❝❤ ❡❧❡♠❡♥t x ♦❢ A t❤❡r❡ ✐s ❛ss♦❝✐❛t❡❞✱ ✐♥ s♦♠❡ ♠❛♥♥❡r✱ ❛♥ ❡❧❡♠❡♥t ♦❢ B✱ ✇❤✐❝❤ ✇❡ ❞❡♥♦t❡ ❜② f(x)✳ ❚❤❡♥ f ✐s s❛✐❞ t♦ ❜❡ ❛ ❢✉♥❝t✐♦♥ ❢r♦♠ A t♦ B ✭♦r ❛ ♠❛♣♣✐♥❣ ♦❢ A ✐♥t♦ B✮✳ ❚❤❡ s❡t A ✐s ❝❛❧❧❡❞ t❤❡ ❞♦♠❛✐♥ ♦❢ f ✭✇❡ ❛❧s♦ s❛② f ✐s ❞❡✜♥❡❞ ♦♥ A✮✱ ❛♥❞ t❤❡ ❡❧❡♠❡♥ts ♦❢ f(x) ❛r❡ ❝❛❧❧❡❞ t❤❡ ✈❛❧✉❡s ♦❢ f. ❚❤❡ s❡t ♦❢ ❛❧❧ ✈❛❧✉❡s ♦❢ f ✐s ❝❛❧❧❡❞ t❤❡ r❛♥❣❡ ♦❢ f.

❉❡✜♥✐t✐♦♥ ✶✼

▲❡t A ❛♥❞ B ❜❡ t✇♦ s❡ts ❛♥❞ ❧❡t f ❜❡ ❛ ♠❛♣♣✐♥❣ ♦❢ A ✐♥t♦ B✳ ■❢ E ⊂ A, f(E) ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ❡❧❡♠❡♥ts f(x), ❢♦r x ∈ E. ❲❡ ❝❛❧❧ f(E) t❤❡ ✐♠❛❣❡ ♦❢ E ✉♥❞❡r f. ■♥ t❤✐s ♥♦t❛t✐♦♥✱ f(A) ✐s t❤❡ r❛♥❣❡ ♦❢ f✳ ■t ✐s ❝❧❡❛r t❤❛t f(A) ⊂ B. ■❢ f(A) = B✱ ✇❡ s❛② t❤❛t f ♠❛♣s A ♦♥t♦ B✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✸✶ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② ❋✐♥✐t❡✱ ❈♦✉♥t❛❜❧❡✱ ❛♥❞ ❯♥❝♦✉♥t❛❜❧❡ ❙❡ts

❉❡✜♥✐t✐♦♥ ✶✽

■❢ E ⊂ B, f−1 ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ❛❧❧ x ∈ A s✉❝❤ t❤❛t f(x) ∈ E. ❲❡ ❝❛❧❧ f−1(E) t❤❡ ✐♥✈❡rs❡ ✐♠❛❣❡ ♦❢ E ✉♥❞❡r f. f ✐s ❛ ✶✲✶ ♠❛♣♣✐♥❣ ♦❢ A ✐♥t♦ B ♣r♦✈✐❞❡❞ t❤❛t f(x1) = f(x2) ✇❤❡♥❡✈❡r x1 = x2, x1 ∈ A, x2 ∈ A.

❉❡✜♥✐t✐♦♥ ✶✾

■❢ t❤❡r❡ ❡①✐sts ❛ ✶✲✶ ♠❛♣♣✐♥❣ ♦❢ A ♦♥t♦ B✱ ✇❡ s❛② t❤❛t A ❛♥❞ B✱ ❝❛♥ ❜❡ ♣✉t ✐♥ ✶✲✶ ❝♦rr❡s♣♦♥❞❡♥❝❡✱ ♦r t❤❛t A ❛♥❞ B ❤❛✈❡ t❤❡ s❛♠❡ ❝❛r❞✐♥❛❧ ♥✉♠❜❡r✱ ♦r A ❛♥❞ B ❛r❡ ❡q✉✐✈❛❧❡♥t✱ ❛♥❞ ✇❡ ✇r✐t❡ A ∼ B✳ Pr♦♣❡rt✐❡s ♦❢ ❡q✉✐✈❛❧❡♥❝❡

◮ ■t ✐s r❡✢❡①✐✈❡✿ A ∼ A. ◮ ■t ✐s s②♠♠❡tr✐❝✿ ■❢ A ∼ B, t❤❡♥ B ∼ A. ◮ ■t ✐s tr❛♥s✐t✐✈❡✿ ■❢ A ∼ B ❛♥❞ B ∼ C, t❤❡♥ A ∼ C. Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✸✷ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② ❋✐♥✐t❡✱ ❈♦✉♥t❛❜❧❡✱ ❛♥❞ ❯♥❝♦✉♥t❛❜❧❡ ❙❡ts

❉❡✜♥✐t✐♦♥ ✷✵

▲❡t n ∈ N ❛♥❞ Jn ❜❡ t❤❡ s❡t ✇❤♦s❡ ❡❧❡♠❡♥ts ❛r❡ t❤❡ ✐♥t❡❣❡rs 1, 2, . . . , n; ❧❡t J ❜❡ t❤❡ s❡t ❝♦♥s✐st✐♥❣ ♦❢ ❛❧❧ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs✳ ❋♦r ❛♥② s❡t A✱ ✇❡ s❛②✿ ✭❛✮ A ✐s ✜♥✐t❡ ✐❢ A ∼ Jn ❢♦r s♦♠❡ n✳ ✭❜✮ A ✐s ✐♥✜♥✐t❡ ✐❢ A ✐s ♥♦t ✜♥✐t❡✳ ✭❝✮ A ✐s ❝♦✉♥t❛❜❧❡ ✐❢ A ∼ J. ✭❞✮ A ✐s ✉♥❝♦✉♥t❛❜❧❡ ✐❢ A ✐s ♥❡✐t❤❡r ✜♥✐t❡ ♥♦r ❝♦✉♥t❛❜❧❡✳ ✭❡✮ A ✐s ❛t ♠♦st ❝♦✉♥t❛❜❧❡ ✐❢ A ✐s ✜♥✐t❡ ♦r ❝♦✉♥t❛❜❧❡✳

❘❡♠❛r❦ ✷

A ✐s ✐♥✜♥✐t❡ ✐❢ A ✐s ❡q✉✐✈❛❧❡♥t t♦ ♦♥❡ ♦❢ ✐ts ♣r♦♣❡r s✉❜s❡ts✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✸✸ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② ❋✐♥✐t❡✱ ❈♦✉♥t❛❜❧❡✱ ❛♥❞ ❯♥❝♦✉♥t❛❜❧❡ ❙❡ts

❉❡✜♥✐t✐♦♥ ✷✶

❇② ❛ s❡q✉❡♥❝❡✱ ✇❡ ♠❡❛♥ ❛ ❢✉♥❝t✐♦♥ f ❞❡✜♥❡❞ ♦♥ t❤❡ s❡t J ♦❢ ❛❧❧ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs✳ ■❢ f(n) = xn✱ ❢♦r n ∈ J✱ ✐t ✐s ❝✉st♦♠❛r② t♦ ❞❡♥♦t❡ t❤❡ s❡q✉❡♥❝❡ f ❜② t❤❡ s②♠❜♦❧ {xn}, ♦r s♦♠❡t✐♠❡s x1, x2, x3, . . . . ❚❤❡ ✈❛❧✉❡s ♦❢ f ❛r❡ ❝❛❧❧❡❞ t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡✳ ■❢ A ✐s ❛ s❡t ❛♥❞ ✐❢ xn ∈ A ❢♦r ❛❧❧ n ∈ J, t❤❡♥ {xn} ✐s s❛✐❞ t♦ ❜❡ ❛ s❡q✉❡♥❝❡ ✐♥ A✱ ♦r ❛ s❡q✉❡♥❝❡ ♦❢ ❡❧❡♠❡♥ts ♦❢ A. ❊✈❡r② ✐♥✜♥✐t❡ s✉❜s❡t ♦❢ ❛ ❝♦✉♥t❛❜❧❡ s❡t A ✐s ❝♦✉♥t❛❜❧❡✳ ❈♦✉♥t❛❜❧❡ s❡ts r❡♣r❡s❡♥t t❤❡ ✏s♠❛❧❧❡st ✐♥✜♥✐t②✳

❉❡✜♥✐t✐♦♥ ✷✷

▲❡t A ❛♥❞ Ω ❜❡ s❡ts✱ ❛♥❞ s✉♣♣♦s❡ t❤❛t ✇✐t❤ ❡❛❝❤ ❡❧❡♠❡♥t ♦❢ α ♦❢ A ✐s ❛ss♦❝✐❛t❡❞ ❛ s✉❜s❡t ♦❢ Ω ✇❤✐❝❤ ❞❡♥♦t❡ ❜② Eα✳ ❆ ❝♦❧❧❡❝t✐♦♥ ♦❢ s❡ts ✐s ❞❡♥♦t❡❞ ❜② {Eα}.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✸✹ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② ❋✐♥✐t❡✱ ❈♦✉♥t❛❜❧❡✱ ❛♥❞ ❯♥❝♦✉♥t❛❜❧❡ ❙❡ts

❉❡✜♥✐t✐♦♥ ✷✸

❚❤❡ ✉♥✐♦♥ ♦❢ t❤❡ s❡ts Eα ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ s❡t S s✉❝❤ t❤❛t x ∈ S ✐❢ ❛♥❞ ♦♥❧② ✐❢ x ∈ Eα ❢♦r ❛t ❧❡❛st ♦♥❡ α ∈ A. ■t ✐s ❞❡♥♦t❡❞ ❜② S =

• α∈A

Eα. ✭✶✷✮ ■❢ A ❝♦♥s✐sts ♦❢ t❤❡ ✐♥t❡❣❡rs 1, 2, . . . , n✱ ♦♥❡ ✉s✉❛❧❧② ✇r✐t❡s S =

n

• m=1

Em = E1 ∪ E2 ∪ · · · ∪ En. ✭✶✸✮ ■❢ A ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs✱ t❤❡ ✉s✉❛❧ ♥♦t❛t✐♦♥s ✐s S =

∞

• m=1

Em. ✭✶✹✮ ❚❤❡ s②♠❜♦❧ ∞ ✐♥❞✐❝❛t❡s t❤❛t t❤❡ ✉♥✐♦♥ ♦❢ ❛ ❝♦✉♥t❛❜❧❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ s❡ts ✐s t❛❦❡♥✳ ■t s❤♦✉❧❞ ♥♦t ❜❡ ❝♦♥❢✉s❡❞ ✇✐t❤ s②♠❜♦❧s +∞ ❛♥❞ −∞ ✐♥tr♦❞✉❝❡❞ ✐♥ ❉❡✜♥✐t✐♦♥ ✶✸✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✸✺ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② ❋✐♥✐t❡✱ ❈♦✉♥t❛❜❧❡✱ ❛♥❞ ❯♥❝♦✉♥t❛❜❧❡ ❙❡ts

❉❡✜♥✐t✐♦♥ ✷✹

❚❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ s❡ts Eα ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ s❡t P s✉❝❤ t❤❛t x ∈ P ✐❢ ❛♥❞ ♦♥❧② ✐❢ x ∈ Eα ❢♦r ❡✈❡r② α ∈ A. ■t ✐s ❞❡♥♦t❡❞ ❜② P =

• α∈A

Eα. ✭✶✺✮ P ✐s ❛❧s♦ ✇r✐tt❡♥ s✉❝❤ ❛s P =

n

• m=1

= E1 ∩ E2 ∩ · · · En. ✭✶✻✮ ■❢ A ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs✱ ✇❡ ❤❛✈❡ P =

∞

• m=1

Em. ✭✶✼✮

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✸✻ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② ❋✐♥✐t❡✱ ❈♦✉♥t❛❜❧❡✱ ❛♥❞ ❯♥❝♦✉♥t❛❜❧❡ ❙❡ts

❚❤❡♦r❡♠ ✻

▲❡t {En}, n = 1, 2, 3, . . . , ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦✉♥t❛❜❧❡ s❡ts✱ ❛♥❞ ♣✉t S =

∞

• n=1

En. ✭✶✽✮ ❚❤❡♥ S ✐s ❝♦✉♥t❛❜❧❡✳ ❚❤❡ s❡t ♦❢ ❛❧❧ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ✐s ❝♦✉♥t❛❜❧❡✳ ❚❤❡ s❡t ♦❢ ❛❧❧ r❡❛❧ ♥✉♠❜❡rs ✐s ✉♥❝♦✉♥t❛❜❧❡✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✸✼ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② ▼❡tr✐❝ ❙♣❛❝❡s

✷✳✷ ▼❡tr✐❝ ❙♣❛❝❡s

❉❡✜♥✐t✐♦♥ ✷✺

❆ s❡t X✱ ✇❤♦s❡ ❡❧❡♠❡♥ts ✇❡ s❤❛❧❧ ❝❛❧❧ ♣♦✐♥ts✱ ✐s s❛✐❞ t♦ ❜❡ ❛ ♠❡tr✐❝ s♣❛❝❡ ✐❢ ✇✐t❤ ❛♥② t✇♦ ♣♦✐♥ts p ❛♥❞ q ♦❢ X t❤❡r❡ ✐s ❛ss♦❝✐❛t❡❞ ❛ r❡❛❧ ♥✉♠❜❡r d(p, q) t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ p t♦ q✱ s✉❝❤ t❤❛t ✭❛✮ d(p, q) > 0 ✐❢ p = q❀ d(p, p) = 0✳ ✭❜✮ d(p, q) = d(q, p); ✭❝✮ d(p, q) ≤ d(p, r) + d(r, q), ❢♦r ❛♥② r ∈ X.

❉❡✜♥✐t✐♦♥ ✷✻

❇② t❤❡ s❡❣♠❡♥t (a, b) ✇❡ ♠❡❛♥ t❤❡ s❡t ♦❢ ❛❧❧ r❡❛❧ ♥✉♠❜❡rs x s✉❝❤ t❤❛t a < x < b.

❉❡✜♥✐t✐♦♥ ✷✼

❇② t❤❡ ✐♥t❡r✈❛❧ [a, b] ✇❡ ♠❡❛♥ t❤❡ s❡t ♦❢ ❛❧❧ r❡❛❧ ♥✉♠❜❡r x s✉❝❤ t❤❛t a ≤ x ≤ b.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✸✽ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② ▼❡tr✐❝ ❙♣❛❝❡s

❉❡✜♥✐t✐♦♥ ✷✽

■❢ x ∈ Rk ❛♥❞ r > 0✱ t❤❡ ♦♣❡♥ ✭♦r ❝❧♦s❡❞✮ ❜❛❧❧ B ✇✐t❤ ❝❡♥t❡r ❛t x ❛♥❞ r❛❞✐✉s r ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ y ∈ Rk s✉❝❤ t❤❛t |y − x| < r ✭♦r |y − x| ≤ r✮✳

❉❡✜♥✐t✐♦♥ ✷✾

❲❡ ❝❛❧❧ ❛ s❡t E ⊂ Rk ❝♦♥✈❡① ✐❢ (λx + (1 − λ)y) ∈ E ✇❤❡♥❡✈❡r x ∈ E✱ y ∈ E ❛♥❞ 0 < λ < 1.

❊①❛♠♣❧❡ ✸

❇❛❧❧s ❛r❡ ❝♦♥✈❡①✳ ❋♦r ✐❢ |y − x| < r✱ |z − x| < r✱ ❛♥❞ 0 < λ < 1, ✇❡ ❤❛✈❡ |λy + (1 − λ)z − x| = |λ(y − x) + (1 − λ)(z − x)| ≤ λ|y − x| + (1 − λ)|z − x| < λr + (1 − λ)r = r.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✸✾ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② ▼❡tr✐❝ ❙♣❛❝❡s

❉❡✜♥✐t✐♦♥ ✸✵

▲❡t X ❜❡ ❛ ♠❡tr✐❝ s♣❛❝❡✳ ❆❧❧ ♣♦✐♥ts ❛♥❞ s❡ts ❛r❡ ❡❧❡♠❡♥ts ❛♥❞ s✉❜s❡ts ♦❢ X. ✭❛✮ ❆ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ❛ ♣♦✐♥t p ✐s ❛ s❡t Nr(p) ❝♦♥s✐st✐♥❣ ♦❢ ❛❧❧ ♣♦✐♥ts q s✉❝❤ t❤❛t d(p, q) < r. ✭❜✮ ❆ ♣♦✐♥t p ✐s ❛ ❧✐♠✐t ♣♦✐♥t ♦❢ t❤❡ s❡t E ✐❢ ❡✈❡r② ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ p ❝♦♥t❛✐♥s ❛ ♣♦✐♥t q = p s✉❝❤ t❤❛t q ∈ E. ✭❝✮ ■❢ p ∈ E ❛♥❞ p ✐s ♥♦t ❛ ❧✐♠✐t ♣♦✐♥t ♦❢ E✱ t❤❡♥ p ✐s ❝❛❧❧❡❞ ❛♥ ✐s♦❧❛t❡❞ ♣♦✐♥t ♦❢ E. ✭❞✮ E ✐s ❝❧♦s❡❞ ✐s ✈❡r② ❧✐♠✐t ♣♦✐♥t ♦❢ E ✐s ❛ ♣♦✐♥t ♦❢ E. ✭❡✮ ❆ ♣♦✐♥t p ✐s ❛♥ ✐♥t❡r✐♦r ♣♦✐♥t ♦❢ E ✐❢ t❤❡r❡ ✐s ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ N ♦❢ p s✉❝❤ t❤❛t N ⊂ E. ✭❢✮ E ✐s ♦♣❡♥ ✐s ❡✈❡r② ♣♦✐♥t ♦❢ E ✐s ❛♥ ✐♥t❡r✐♦r ♣♦✐♥t ♦❢ E. ✭❣✮ ❚❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ E ✭❞❡♥♦t❡❞ ❜② Ec✮ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦✐♥ts p ∈ X s✉❝❤ t❤❛t p / ∈ E.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✹✵ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② ▼❡tr✐❝ ❙♣❛❝❡s

❉❡✜♥✐t✐♦♥ ✸✵

✭❤✮ E ✐s ♣❡r❢❡❝t ✐❢ E ✐s ❝❧♦s❡❞ ❛♥❞ ✐❢ ❡✈❡r② ♣♦✐♥t ♦❢ E ✐s ❛ ❧✐♠✐t ♣♦✐♥t ♦❢ E. ✭✐✮ E ✐s ❜♦✉♥❞❡❞ ✐❢ t❤❡r❡ ✐s ❛ r❡❛❧ ♥✉♠❜❡r M ❛♥❞ ❛ ♣♦✐♥t q ∈ X s✉❝❤ t❤❛t d(p, q) < M ❢♦r ❛❧❧ p ∈ E. ✭❥✮ E ✐s ❞❡♥s❡ ✐♥ X ✐❢ ❡✈❡r② ♣♦✐♥t ♦❢ X ✐s ❛ ❧✐♠✐t ♣♦✐♥t ♦❢ E✱ ♦r ❛ ♣♦✐♥t ♦❢ E ✭♦r ❜♦t❤✮✳ ■❢ p ✐s ❛ ❧✐♠✐t ♣♦✐♥t ♦❢ ❛ s❡t E✱ t❤❡♥ ❡✈❡r② ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ p ❝♦♥t❛✐♥s ✐♥✜♥✐t❡❧② ♠❛♥② ♣♦✐♥ts ♦❢ E. ❆ s❡t E ✐s ♦♣❡♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❝♦♠♣❧❡♠❡♥t ✐s ❝❧♦s❡❞✳

❉❡✜♥✐t✐♦♥ ✸✶

■❢ X ✐s ❛ ♠❡tr✐❝ s♣❛❝❡✱ ✐❢ E ⊂ X, ❛♥❞ ✐❢ E′ ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ❛❧❧ ❧✐♠✐t ♣♦✐♥ts ♦❢ E ✐♥ X✱ t❤❡♥ t❤❡ ❝❧♦s✉r❡ ♦❢ E ✐s t❤❡ s❡t ¯ E = E ∪ E′.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✹✶ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② ▼❡tr✐❝ ❙♣❛❝❡s

❚❤❡♦r❡♠ ✼

■❢ X ✐s ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ E ⊂ X, t❤❡♥ ✭❛✮ ¯ E ✐s ❝❧♦s❡❞✳ ✭❜✮ E = ¯ E ✐❢ ❛♥❞ ♦♥❧② ✐❢ E ✐s ❝❧♦s❡❞✳ ✭❝✮ E ⊂ F ❢♦r ❡✈❡r② ❝❧♦s❡❞ s❡t F ⊂ X s✉❝❤ t❤❛t E ⊂ F.

❚❤❡♦r❡♠ ✽

▲❡t E ❜❡ ❛ ♥♦♥❡♠♣t② s❡t ♦❢ r❡❛❧ ♥✉♠❜❡rs ✇❤✐❝❤ ✐s ❜♦✉♥❞❡❞ ❛❜♦✈❡✳ ▲❡t y = sup E. ❚❤❡♥ y ∈ ¯ E✳ ❍❡♥❝❡ y ∈ E ✐❢ E ✐s ❝❧♦s❡❞✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✹✷ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② ❈♦♠♣❛❝t ❙❡ts

✷✳✸ ❈♦♠♣❛❝t ❙❡ts

❉❡✜♥✐t✐♦♥ ✸✷

❇② ❛♥ ♦♣❡♥ ❝♦✈❡r ♦❢ ❛ s❡t E ✐♥ ❛ ♠❡tr✐❝ s♣❛❝❡ X ✇❡ ♠❡❛♥ ❛ ❝♦❧❧❡❝t✐♦♥ {Gα} ♦❢ ♦♣❡♥ s✉❜s❡ts ♦❢ X s✉❝❤ t❤❛t E ⊂

α Gα.

❉❡✜♥✐t✐♦♥ ✸✸

❆ s✉❜s❡t K ♦❢ ❛ ♠❡tr✐❝ s♣❛❝❡ X ✐s s❛✐❞ t♦ ❜❡ ❝♦♠♣❛❝t ✐❢ ❡✈❡r② ♦♣❡♥ ❝♦✈❡r ♦❢ K ❝♦♥t❛✐♥s ❛ ✜♥✐t❡ s✉❜❝♦✈❡r✳

❉❡✜♥✐t✐♦♥ ✸✹

❆ s❡t X ⊂ R ✐s ❝♦♠♣❛❝t ✐❢ X ✐s ❝❧♦s❡❞ ❛♥❞ ❜♦✉♥❞❡❞❛✳

❛▲✐♠❛✱ ❊✳ ▲✳ ✭✷✵✵✻✮ ❆♥á❧✐s❡ ❘❡❛❧ ✈♦❧✉♠❡ ✶✳ ❋✉♥çõ❡s ❞❡ ❯♠❛ ❱❛r✐á✈❡❧✳

❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ■▼P❆✱ ✷✵✵✻✳

❉❡✜♥✐t✐♦♥ ✸✺

■❢ {Kn} ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ♥♦♥❡♠♣t② ❝♦♠♣❛❝t s❡ts s✉❝❤ t❤❛t Kn ⊃ Kn+1 (n = 1, 2, 3 . . .),✱ t❤❡♥ ∞

1 Kn ✐s ♥♦t ❡♠♣t②✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✹✸ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② ❈♦♠♣❛❝t ❙❡ts

❉❡✜♥✐t✐♦♥ ✸✻

■❢ {In} ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡r✈❛❧s ✐♥ R1✱ s✉❝❤ t❤❛t In ⊃ In+1 (n = 1, 2, 3 . . .),✱ t❤❡♥ ∞

1 In ✐s ♥♦t ❡♠♣t②✳

❚❤❡♦r❡♠ ✾

■❢ ❛ s❡t E ✐♥ Rk ❤❛s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ♣r♦♣❡rt✐❡s✱ t❤❡♥ ✐t ❤❛s t❤❡ ♦t❤❡r t✇♦✿

✶ E ✐s ❝❧♦s❡❞ ❛♥❞ ❜♦✉♥❞❡❞✳ ✷ E ✐s ❝♦♠♣❛❝t✳ ✸ ❊✈❡r② ✐♥✜♥✐t❡ s✉❜s❡t ♦❢ E ❤❛s ❛ ❧✐♠✐t ♣♦✐♥t ✐♥ E.

❚❤❡♦r❡♠ ✶✵

✭❲❡✐❡rstr❛ss✮ ❊✈❡r② ❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ Rk ❤❛s ❛ ❧✐♠✐t ♣♦✐♥t ✐♥ Rk.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✹✹ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② P❡r❢❡❝t ❙❡ts

✷✳✹ P❡r❢❡❝t ❙❡ts

❚❤❡♦r❡♠ ✶✶

▲❡t P ❜❡ ❛ ♥♦♥❡♠♣t② ♣❡r❢❡❝t s❡t ✐♥ Rk. ❚❤❡♥ P ✐s ✉♥❝♦✉♥t❛❜❧❡✳ ❊✈❡r② ✐♥t❡r✈❛❧ [a, b](a < b) ✐s ✉♥❝♦✉♥t❛❜❧❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ s❡t ♦❢ ❛❧❧ r❡❛❧ ♥✉♠❜❡rs ✐♥ ✉♥❝♦✉♥t❛❜❧❡✳ ❚❤❡ ❈❛♥t♦r t❡r♥❛r② s❡t ✐s ❝r❡❛t❡❞ ❜② r❡♣❡❛t❡❞❧② ❞❡❧❡t✐♥❣ t❤❡ ♦♣❡♥ ♠✐❞❞❧❡ t❤✐r❞s ♦❢ ❛ s❡t ♦❢ ❧✐♥❡ s❡❣♠❡♥ts✳ ❖♥❡ st❛rts ❜② ❞❡❧❡t✐♥❣ t❤❡ ♦♣❡♥ ♠✐❞❞❧❡ t❤✐r❞ (1/3, 2/3) ❢r♦♠ t❤❡ ✐♥t❡r✈❛❧ [0, 1], ❧❡❛✈✐♥❣ t✇♦ ❧✐♥❡ s❡❣♠❡♥ts✿ [0, 1/3] ∪ [2/3, 1] ✳ ◆❡①t✱ t❤❡ ♦♣❡♥ ♠✐❞❞❧❡ t❤✐r❞ ♦❢ ❡❛❝❤ ♦❢ t❤❡s❡ r❡♠❛✐♥✐♥❣ s❡❣♠❡♥ts ✐s ❞❡❧❡t❡❞✱ ❧❡❛✈✐♥❣ ❢♦✉r ❧✐♥❡ s❡❣♠❡♥ts✿ [0, 1/9] ∪ [2/9, 1/3] ∪ [2/3, 7/9] ∪ [8/9, 1] ✳ ❚❤✐s ♣r♦❝❡ss ✐s ❝♦♥t✐♥✉❡❞ ❛❞ ✐♥✜♥✐t✉♠✱ ✇❤❡r❡ t❤❡ ♥t❤ s❡t ✐s Cn = Cn−1 3 ∪ 2 3 + Cn−1 3

• .C0 = [0, 1].

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✹✺ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② P❡r❢❡❝t ❙❡ts

❚❤❡ ✜rst s✐① st❡♣s ♦❢ t❤✐s ♣r♦❝❡ss ❛r❡ ✐❧❧✉str❛t❡❞ ✐♥ ❋✐❣✉r❡ ✹✻✳

❋✐❣✉r❡ ✷✿ ❈❛♥t♦r ❙❡t✳ ❙♦✉r❝❡✿ ❲✐❦✐♣❡❞✐❛✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✹✻ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② ❈♦♥♥❡❝t❡❞ ❙❡ts

✷✳✺ ❈♦♥♥❡❝t❡❞ ❙❡ts

❉❡✜♥✐t✐♦♥ ✸✼

❚✇♦ s✉❜s❡ts A ❛♥❞ B ♦❢ ❛ ♠❡tr✐❝ s♣❛❝❡ X ❛r❡ s❛✐❞ t♦ ❜❡ s❡♣❛r❛t❡❞ ✐❢ ❜♦t❤ A ∩ ¯ B ❛♥❞ ¯ A ∩ B ❛r❡ ❡♠♣t②✱ ✐✳❡✳✱ ✐❢ ♥♦ ♣♦✐♥t ♦❢ A ❧✐❡s ✐♥ t❤❡ ❝❧♦s✉r❡ ♦❢ B ❛♥❞ ♥♦ ♣♦✐♥t ♦❢ B ❧✐❡s ✐♥ t❤❡ ❝❧♦s✉r❡ ♦❢ A. ❆ s❡t E ⊂ X ✐s s❛✐❞ t♦ ❜❡ ❝♦♥♥❡❝t❡❞ ✐❢ E ✐s ♥♦t ❛ ✉♥✐♦♥ ♦❢ t✇♦ ♥♦♥❡♠♣t② s❡♣❛r❛t❡❞ s❡ts✳

❚❤❡♦r❡♠ ✶✷

❆ s✉❜s❡t E ♦❢ t❤❡ r❡❛❧ ❧✐♥❡ R1 ✐s ❝♦♥♥❡❝t❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt②✿ ■❢ x ∈ E✱ y ∈ E, ❛♥❞ x < z < y, t❤❡♥ z ∈ E.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✹✼ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② ❈♦♥♥❡❝t❡❞ ❙❡ts

❊①❡r❝✐s❡s ❈❤❛♣t❡r ✷

✭✶✮ ▲❡t A ❜❡ t❤❡ s❡t ♦❢ r❡❛❧ ♥✉♠❜❡rs x s✉❝❤ t❤❛t 0 < x ≤ 1. ❋♦r ❡✈❡r② x ∈ A✱ ❜❡ t❤❡ s❡t ♦❢ r❡❛❧ ♥✉♠❜❡rs y✱ s✉❝❤ t❤❛t 0 < y < x. ❈♦♠♣❧❡t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ✭❛✮ Ex ⊂ Ez ✐❢ ❛♥❞ ♦♥❧② ✐❢ 0 < x ≤ z ≤ 1✳ ✭❜✮

x∈A Ex = E1✳

✭❝✮

x∈A Ex ✐s ❡♠♣t②✳

✭✷✮ Pr♦✈❡ ❚❤❡♦r❡♠ ✻✳ ❍✐♥t✿ ♣✉t t❤❡ ❡❧❡♠❡♥ts ♦❢ En ✐♥ ❛ ♠❛tr✐① ❛♥❞ ❝♦✉♥t t❤❡ ❞✐❛❣♦♥❛❧s✳ ✭✸✮ Pr♦✈❡ t❤❛t t❤❡ s❡t ♦❢ ❛❧❧ r❡❛❧ ♥✉♠❜❡rs ✐s ✉♥❝♦✉♥t❛❜❧❡✳ ✭✹✮ ❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❡①❛♠♣❧❡s ♦❢ ♠❡tr✐❝ s♣❛❝❡s ❛r❡ ❡✉❝❧✐❞❡❛♥ s♣❛❝❡s Rk. ❙❤♦✇ t❤❛t ❛ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✐s ❛ ♠❡tr✐❝ s♣❛❝❡✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✹✽ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② ❈♦♥♥❡❝t❡❞ ❙❡ts

✭✺✮ ❋♦r x ∈ R1 ❛♥❞ y ∈ R1✱ ❞❡✜♥❡ d1(x, y) = (x − y)2, d2(x, y) =

• |x − y|,

d3(x, y) = |x2 − y2|, d4(x, y) = |x − 2y|, d5(x, y) = |x − y| 1 + |x − y|. ❉❡t❡r♠✐♥❡ ❢♦r ❡❛❝❤ ♦❢ t❤❡s❡✱ ✇❤❡t❤❡r ✐t ✐s ❛ ♠❡tr✐❝ ♦r ♥♦t✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✹✾ ✴ ✽✶

❇❛s✐❝ ❚♦♣♦❧♦❣② ❈♦♥♥❡❝t❡❞ ❙❡ts

❲♦r❦ ✶

❚♦ ✜♥❞ t❤❡ sq✉❛r❡ r♦♦t ♦❢ ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r a✱ ✇❡ st❛rt ✇✐t❤ s♦♠❡ ❛♣♣r♦①✐♠❛t✐♦♥✱ x0 > 0 ❛♥❞ t❤❡♥ r❡❝✉rs✐✈❡❧② ❞❡✜♥❡✿ xn+1 = 1 2

• xn + a

xn

• .

✭✶✾✮ ❈♦♠♣✉t❡ t❤❡ sq✉❛r❡ r♦♦t ✉s✐♥❣ ✭✶✾✮ ❢♦r ✭❛✮ a = 2; ✭❜✮ a = 2 × 10−300 ✭❝✮ a = 2 × 10−310 ✭❞✮ a = 2 × 10−322 ✭❡✮ a = 2 × 10−324 ❈❤❡❝❦ ②♦✉r r❡s✉❧ts ❜② xn × xn✱ ❛❢t❡r ❞❡✜♥✐♥❣ ❛ s✉✐t❛❜❧❡ st♦♣ ❝r✐t❡r✐❛ ❢♦r n✳ ❉❡✈❡❧♦♣ ❛ r❡♣♦rt ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ str✉❝t✉r❡✿ ■❞❡♥t✐✜❝❛t✐♦♥✱ ■♥tr♦❞✉❝t✐♦♥✱ ▼❡t❤♦❞♦❧♦❣②✱ ❘❡s✉❧ts✱ ❈♦♥❝❧✉s✐♦♥✱ ❘❡❢❡r❡♥❝❡s✱ ❆♣♣❡♥❞✐① ✭✇❤❡r❡ ②♦✉ s❤♦✉❧❞ ✐♥❝❧✉❞❡ ❛♥ ❛❧❣♦r✐t❤♠✮✳ ❉❡❛❞❧✐♥❡✿ ✶✵✴✵✾✴✷✵✶✹✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✺✵ ✴ ✽✶

◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ❈♦♥✈❡r❣❡♥t ❙❡q✉❡♥❝❡s

✸✳ ◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s

✸✳✶ ❈♦♥✈❡r❣❡♥t ❙❡q✉❡♥❝❡s

❉❡✜♥✐t✐♦♥ ✸✽

❆ s❡q✉❡♥❝❡ {pn} ✐♥ ❛ ♠❡tr✐❝ s♣❛❝❡ X ✐s s❛✐❞ t♦ ❝♦♥✈❡r❣❡ ✐❢ t❤❡r❡ ✐s ♣♦✐♥t p ∈ X ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt②✿ ❋♦r ❡✈❡r② ε > 0 t❤❡r❡ ✐s ❛♥ ✐♥t❡❣❡r N s✉❝❤ t❤❛t n ≥ N ✐♠♣❧✐❡s t❤❛t d(pn, p) < ε✳ ■♥ t❤✐s ❝❛s❡ ✇❡ ❛❧s♦ s❛② t❤❛t pn ❝♦♥✈❡r❣❡s t♦ p, ♦r t❤❛t p ✐s t❤❡ ❧✐♠✐t ♦❢ {pn}✱ ❛♥❞ ✇❡ ✇r✐t❡ pn → p, ♦r lim

n→∞ pn = p.

■❢ {pn} ❞♦❡s ♥♦t ❝♦♥✈❡r❣❡✱ ✐t ✐s s❛✐❞ t♦ ❞✐✈❡r❣❡✳ ■t ♠✐❣❤t ❜❡ ✇❡❧❧ t♦ ♣♦✐♥t ♦✉t t❤❛t ♦✉r ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ❞❡♣❡♥❞s ♥♦t ♦♥❧② ♦♥ {pn} ❜✉t ❛❧s♦ ♦♥ X. ■t ✐s ♠♦r❡ ♣r❡❝✐s❡ t♦ s❛② ❝♦♥✈❡r❣❡♥t ✐♥ X. ❚❤❡ s❡t ♦❢ ❛❧❧ ♣♦✐♥ts pn (n = 1, 2, 3, . . .) ✐s t❤❡ r❛♥❣❡ ♦❢ {pn}✳ ❚❤❡ s❡q✉❡♥❝❡ {pn} ✐s s❛✐❞ t♦ ❜❡ ❜♦✉♥❞❡❞ ✐❢ ✐ts r❛♥❣❡ ✐s ❜♦✉♥❞❡❞✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✺✶ ✴ ✽✶

◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ❈♦♥✈❡r❣❡♥t ❙❡q✉❡♥❝❡s

❊①❛♠♣❧❡ ✹

▲❡t s ∈ R. ■❢ sn = 1/n✱ t❤❡♥ lim

n→∞ sn = 0.

❚❤❡ r❛♥❣❡ ✐s ✐♥✜♥✐t❡✱ ❛♥❞ t❤❡ s❡q✉❡♥❝❡ ✐s ❜♦✉♥❞❡❞✳

❊①❛♠♣❧❡ ✺

▲❡t s ∈ R. ■❢ sn = n2✱ t❤❡ s❡q✉❡♥❝❡ {sn} ✐s ✉♥❜♦✉♥❞❡❞✱ ✐s ❞✐✈❡r❣❡♥t✱ ❛♥❞ ❤❛s ✐♥✜♥✐t❡ r❛♥❣❡✳

❊①❛♠♣❧❡ ✻

▲❡t s ∈ R. ■❢ sn = 1 (n = 1, 2, 3, . . .)✱ t❤❡♥ t❤❡ s❡q✉❡♥❝❡ {sn} ❝♦♥✈❡r❣❡s t♦ ✶✱ ✐s ❜♦✉♥❞❡❞✱ ❛♥❞ ❤❛s ✜♥✐t❡ r❛♥❣❡✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✺✷ ✴ ✽✶

◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ❈♦♥✈❡r❣❡♥t ❙❡q✉❡♥❝❡s

❚❤❡♦r❡♠ ✶✸

▲❡t {pn} ❜❡ ❛ s❡q✉❡♥❝❡ ✐♥ ❛ ♠❡tr✐❝ s♣❛❝❡ X. ✭❛✮ {pn} ❝♦♥✈❡r❣❡s t♦ p ∈ X ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✈❡r② ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ p ❝♦♥t❛✐♥s ❛❧❧ ❜✉t ✜♥✐t❡❧② ♠❛♥② ♦❢ t❤❡ t❡r♠s ♦❢ {pn}✳ ✭❜✮ ■❢ p ∈ X, p′ ∈ X, ❛♥❞ ✐❢ {pn} ❝♦♥✈❡r❣❡s t♦ p ❛♥❞ t♦ p′ ✱ t❤❡♥ p′ = p. ✭❝✮ ■❢ {pn} ❝♦♥✈❡r❣❡s✱ t❤❡♥ {pn} ✐s ❜♦✉♥❞❡❞✳ ✭❞✮ ■❢ E ⊂ X ❛♥❞ ✐❢ p ✐s ❛ ❧✐♠✐t ♣♦✐♥t ♦❢ E✱ t❤❡♥ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ {pn} ✐♥ E s✉❝❤ t❤❛t p = lim

n→∞ pn.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✺✸ ✴ ✽✶

◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ❙✉❜s❡q✉❡♥❝❡s

❚❤❡♦r❡♠ ✶✹

❙✉♣♣♦s❡ {sn}✱ {tn} ❛r❡ ❝♦♠♣❧❡① s❡q✉❡♥❝❡s✱ ❛♥❞ limn→∞sn = s ❛♥❞ limn→∞tn = t. ❚❤❡♥ ✭❛✮ lim

n→∞(sn + tn) = s + t;

✭❜✮ lim

n→∞ csn = cs, lim n→∞(c + sn) = c + s, ❢♦r ❛♥② ♥✉♠❜❡r c;

✭❝✮ lim

n→∞(sntn) = st;

✭❞✮ lim

n→∞

1 sn = 1 s; ✸✳✷ ❙✉❜s❡q✉❡♥❝❡s

❉❡✜♥✐t✐♦♥ ✸✾

• ✐✈❡♥ ❛ s❡q✉❡♥❝❡ {pn}✱ ❝♦♥s✐❞❡r ❛ s❡q✉❡♥❝❡ {nk} ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs✱

s✉❝❤ t❤❛t n1 < n2 < n3 < · · · . ❚❤❡♥ t❤❡ s❡q✉❡♥❝❡ {pni} ✐s ❝❛❧❧❡❞ ❛ s✉❜s❡q✉❡♥❝❡ ♦❢ {pn}✳ ■❢ {pni}✱ ✐ts ❧✐♠✐t ✐s ❝❛❧❧❡❞ ❛ s✉❜s❡q✉❡♥t✐❛❧ ❧✐♠✐t ♦❢ {pn}✳ ■t ✐s ❝❧❡❛r t❤❛t {pn} ❝♦♥✈❡r❣❡s t♦ p ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✈❡r② s✉❜s❡q✉❡♥❝❡ ♦❢ {pn} ❝♦♥✈❡r❣❡s t♦ p.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✺✹ ✴ ✽✶

◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ❙✉❜s❡q✉❡♥❝❡s

❚❤❡♦r❡♠ ✶✺

✭❛✮ ■❢ {pn} ✐s ❛ s❡q✉❡♥❝❡ ✐♥ ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ X, t❤❡♥ s♦♠❡ s✉❜s❡q✉❡♥❝❡ ♦❢ {pn} ❝♦♥✈❡r❣❡s t♦ ❛ ♣♦✐♥t ♦❢ X. ✭❜✮ ❊✈❡r② ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ✐♥ Rk ❝♦♥t❛✐♥s ❛ ❝♦♥✈❡r❣❡♥t s✉❜s❡q✉❡♥❝❡✳

❚❤❡♦r❡♠ ✶✻

❚❤❡ s✉❜s❡q✉❡♥t✐❛❧ ❧✐♠✐ts ♦❢ ❛ s❡q✉❡♥❝❡ {pn} ✐♥ ❛ ♠❡tr✐❝ s♣❛❝❡X ❢♦r♠ ❛ ❝❧♦s❡❞ s✉❜s❡t ♦❢ X.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✺✺ ✴ ✽✶

◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ❈❛✉❝❤② ❙❡q✉❡♥❝❡

✸✳✸ ❈❛✉❝❤② ❙❡q✉❡♥❝❡

❉❡✜♥✐t✐♦♥ ✹✵

❆ s❡q✉❡♥❝❡ {pn} ✐s ❛ ♠❡tr✐❝ s♣❛❝❡ X ✐s s❛✐❞ t♦ ❜❡ ❛ ❈❛✉❝❤② s❡q✉❡♥❝❡ ✐❢ ❢♦r ❡✈❡r② ε > 0 t❤❡r❡ ✐s ❛♥ ✐♥t❡❣❡r N s✉❝❤ t❤❛t d(pn, pm) < ε ✐❢ n ≥ N ❛♥❞ m ≥ N.

❋✐❣✉r❡ ✸✿ ❆✉❣✉st✐♥✲▲♦✉✐s ❈❛✉❝❤② ✭✶✼✽✾✲✶✽✺✼✮✱ ❋r❡♥❝❤ ♠❛t❤❡♠❛t✐❝✐❛♥ ✇❤♦ ✇❛s ❛♥ ❡❛r❧② ♣✐♦♥❡❡r ♦❢ ❛♥❛❧②s✐s✳ ❙♦✉r❝❡✿ ❲✐❦✐♣❡❞✐❛✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✺✻ ✴ ✽✶

◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ❈❛✉❝❤② ❙❡q✉❡♥❝❡

❉❡✜♥✐t✐♦♥ ✹✶

▲❡t E ❜❡ ❛ s✉❜s❡t ♦❢ ❛ ♠❡tr✐❝ s♣❛❝❡ X, ❛♥❞ ❧❡t S ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ r❡❛❧ ♥✉♠❜❡r ♦❢ t❤❡ ❢♦r♠ d(p, q), ✇✐t❤ p ∈ E ❛♥❞ q ∈ E✳ ❚❤❡ s✉♣ ♦❢ S ✐s ❝❛❧❧❡❞ t❤❡ ❞✐❛♠❡t❡r ♦❢ E. ■❢ {pn} ✐s ❛ s❡q✉❡♥❝❡ ✐♥ X ❛♥❞ ✐❢ EN ❝♦♥s✐sts ♦❢ t❤❡ ♣♦✐♥ts pN, pN+1, pN+2, . . . , ✐t ✐s ❝❧❡❛r ❢r♦♠ t❤❡ t✇♦ ♣r❡❝❡❞✐♥❣ ❞❡✜♥✐t✐♦♥s t❤❛t {pn} ✐s ❛ ❈❛✉❝❤② s❡q✉❡♥❝❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ lim

N→∞ diam EN = 0.

❚❤❡♦r❡♠ ✶✼

✭❛✮ ■❢ ¯ E ✐s t❤❡ ❝❧♦s✉r❡ ♦❢ ❛ s❡t E ✐♥ ❛ ♠❡tr✐❝ s♣❛❝❡ X, t❤❡♥ diam ¯ E = diam E. ✭❜✮ ■❢ Ka ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣❛❝t s❡ts ✐♥ X s✉❝❤ t❤❛t Kn ⊃ Kn+1 (n = 1, 2, 3, . . .) ❛♥❞ ✐❢ lim diam Kn = 0, t❤❡♥ ❝♦♥s✐sts ♦❢ ❡①❛❝t❧② ♦♥❡ ♣♦✐♥t✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✺✼ ✴ ✽✶

◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ❈❛✉❝❤② ❙❡q✉❡♥❝❡

❚❤❡♦r❡♠ ✶✽

✭❛✮ ■♥ ❛♥② ♠❡tr✐❝ s♣❛❝❡ X, ❡✈❡r② ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ✐s ❛ ❈❛✉❝❤② s❡q✉❡♥❝❡✳ ✭❜✮ ■❢ X ✐s ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ ✐❢ {pn} ✐s ❛ ❈❛✉❝❤② s❡q✉❡♥❝❡ ✐♥ X, t❤❡♥ {pn} ❝♦♥✈❡r❣❡s t♦ s♦♠❡ ♣♦✐♥t X. ✭❝✮ ■♥ Rk✱ ❡✈❡r② ❈❛✉❝❤② s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡s✳ ❆ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡s ✐♥ Rk ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❛ ❈❛✉❝❤② s❡q✉❡♥❝❡ ✐s ✉s✉❛❧❧② ❝❛❧❧❡❞ t❤❡ ❈❛✉❝❤② ❝r✐t❡r✐♦♥ ❢♦r ❝♦♥✈❡r❣❡♥❝❡✳

❉❡✜♥✐t✐♦♥ ✹✷

❆ s❡q✉❡♥❝❡ {sn} ♦❢ r❡❛❧ ♥✉♠❜❡rs ✐s s❛✐❞ t♦ ❜❡ ✭❛✮ ♠♦♥♦t♦♥✐❝❛❧❧② ✐♥❝r❡❛s✐♥❣ ✐❢ sn ≤ sn+1 (n = 1, 2, 3, . . .); ✭❜✮ ♠♦♥♦t♦♥✐❝❛❧❧② ❞❡❝r❡❛s✐♥❣ ✐❢ sn ≥ sn+1 (n = 1, 2, 3, . . .);

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✺✽ ✴ ✽✶

◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ❯♣♣❡r ❛♥❞ ▲♦✇❡r ▲✐♠✐ts

✸✳✹ ❯♣♣❡r ❛♥❞ ▲♦✇❡r ▲✐♠✐ts

❚❤❡♦r❡♠ ✶✾

❙✉♣♣♦s❡ {sn} ✐s ♠♦♥♦t♦♥✐❝✳ ❚❤❡♥ {sn} ❝♦♥✈❡r❣❡s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❜♦✉♥❞❡❞✳

❉❡✜♥✐t✐♦♥ ✹✸

▲❡t {sn} ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ r❡❛❧ ♥✉♠❜❡rs ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt②✿ ❋♦r ❡✈❡r② r❡❛❧ M t❤❡r❡ ✐s ❛♥ ✐♥t❡❣❡r N s✉❝❤ t❤❛t n ≥ N ✐♠♣❧✐❡s sn ≥ M. ❲❡ t❤❡♥ ✇r✐t❡ sn → +∞✳

❉❡✜♥✐t✐♦♥ ✹✹

▲❡t {sn} ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ r❡❛❧ ♥✉♠❜❡rs✳ ▲❡t E ❜❡ t❤❡ s❡t ♦❢ ♥✉♠❜❡rs x ∈ ¯ R s✉❝❤ t❤❛t snk → x ❢♦r s♦♠❡ s✉❜s❡q✉❡♥❝❡ {snk}✳ ❚❤✐s s❡t E ❝♦♥t❛✐♥s ❛❧❧ s✉❜s❡q✉❡♥t✐❛❧ ❧✐♠✐ts ♣❧✉s ♣♦ss✐❜❧② t❤❡ ♥✉♠❜❡rs +∞ ❛♥❞ −∞. ▲❡t s∗ = sup E, ❛♥❞ s∗ = inf E. ❚❤❡s❡ ♥✉♠❜❡rs ❛r❡ ❝❛❧❧❡❞ ✉♣♣❡r ❛♥❞ ❧♦✇❡r ❧✐♠✐ts ♦❢ {sn}✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✺✾ ✴ ✽✶

◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ❙♦♠❡ ❙♣❡❝✐❛❧ ❙❡q✉❡♥❝❡s

❲❡ ❝❛♥ ❛❧s♦ ✇r✐t❡ ❉❡✜♥✐t✐♦♥ ✹✹ ❛s lim

n→∞ sup sn = s∗,

lim

n→∞ inf sn = s∗.

✸✳✺ ❙♦♠❡ ❙♣❡❝✐❛❧ ❙❡q✉❡♥❝❡s ■❢ 0 ≤ xn ≤ sn ❢♦r n ≥ N, ✇❤❡r❡ N ✐s s♦♠❡ ✜①❡❞ ♥✉♠❜❡r✱ ❛♥❞ ✐❢ sn → 0✱ t❤❡♥ xn → 0✳ ❚❤✐s ♣r♦♣❡rt② ❤❡❧♣ ✉s t♦ ❝♦♠♣✉t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡s✿ ✭❛✮ ■❢ p > 0, t❤❡♥ lim

n→∞

1 np = 0. ✭❜✮ ■❢ p > 0, t❤❡♥ lim

n→∞

n

√p = 1. ✭❝✮ lim

n→∞

n

√n = 1. ✭❞✮ ■❢ p > 0 ❛♥❞ α ✐s r❡❛❧✱ t❤❡♥ lim

n→∞

nα (1 + p)n = 0. ✭❡✮ ■❢ |x| < 1, t❤❡♥ lim

n→∞ xn = 0.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✻✵ ✴ ✽✶

◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ❙❡r✐❡s

✸✳✻ ❙❡r✐❡s

❉❡✜♥✐t✐♦♥ ✹✺

• ✐✈❡♥ ❛ s❡q✉❡♥❝❡ {an}✱ ✇❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥

q

• n=p

an (p ≤ q) t♦ ❞❡♥♦t❡ t❤❡ s✉♠ ap + ap+1 + · · · + aq. ❲✐t❤ {an} ✇❡ ❛ss♦❝✐❛t❡ ❛ s❡q✉❡♥❝❡ {sn}, ✇❤❡r❡ sn =

n

• k=1

ak. ❋♦r {sn} ✇❡ ❛❧s♦ ✉s❡ t❤❡ s②♠❜♦❧✐❝ ❡①♣r❡ss✐♦♥ a1 + a2 + a3 + · · · ♦r✱ ♠♦r❡ ❝♦♥❝✐s❡❧②✱

∞

• n=1

an. ✭✷✵✮ ❚❤❡ s②♠❜♦❧ ✭✸✸✮ ✇❡ ❝❛❧❧ ❛♥ ✐♥✜♥✐t❡ s❡r✐❡s✱ ♦r ❥✉st ❛ s❡r✐❡s✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✻✶ ✴ ✽✶

◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ❙❡r✐❡s

❚❤❡ ♥✉♠❜❡rs sn ❛r❡ ❝❛❧❧❡❞ t❤❡ ♣❛rt✐❛❧ s✉♠s ♦❢ t❤❡ s❡r✐❡s✳ ■❢ {sn} ❝♦♥✈❡r❣❡s t♦ s✱ ✇❡ s❛② t❤❛t t❤❡ s❡r✐❡s ❝♦♥✈❡r❣❡s✱ ❛♥❞ ✇❡ ✇r✐t❡

∞

• n=1

an = s. ✭✷✶✮ s ✐s t❤❡ ❧✐♠✐t ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ s✉♠s✱ ❛♥❞ ✐s ♥♦t ♦❜t❛✐♥❡❞ s✐♠♣❧② ❜② ❛❞❞✐t✐♦♥✳ ■❢ {sn} ❞✐✈❡r❣❡s✱ t❤❡ s❡r✐❡s ✐s s❛✐❞ t♦ ❞✐✈❡r❣❡✳ ❊✈❡r② t❤❡♦r❡♠ ❛❜♦✉t s❡q✉❡♥❝❡s ❝❛♥ ❜❡ st❛t❡❞ ✐♥ t❡r♠s ♦❢ s❡r✐❡s ✭♣✉tt✐♥❣ a1 = s1✱ ❛♥❞ an = sn − sn−1 ❢♦r n > 1✮✱ ❛♥❞ ✈✐❝❡ ✈❡rs❛✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✻✷ ✴ ✽✶

◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ❙❡r✐❡s

❚❤❡ ❈❛✉❝❤② ❝r✐t❡r✐♦♥ ❝❛♥ ❜❡ r❡st❛t❡❞ ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❚❤❡♦r❡♠✳

❚❤❡♦r❡♠ ✷✵

an ❝♦♥✈❡r❣❡s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❢♦r ❡✈❡r② ε > 0 t❤❡r❡ ✐s ❛♥ ✐♥t❡❣❡r N s✉❝❤ t❤❛t

• m
• k=n

an

• ≤ ε

✭✷✷✮ ✐❢ m ≥ n ≥ N.

❚❤❡♦r❡♠ ✷✶

■❢ an ❝♦♥✈❡r❣❡s✱ t❤❡♥ lim

n→∞ an = 0.

❚❤❡♦r❡♠ ✷✷

❆ s❡r✐❡s ♦❢ ♥♦♥♥❡❣❛t✐✈❡ t❡r♠s ❝♦♥✈❡r❣❡s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ♣❛rt✐❛❧ s✉♠s ❢♦r♠ ❛ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✻✸ ✴ ✽✶

◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ❙❡r✐❡s

❈♦♠♣❛r✐s♦♥ t❡st ✭❛✮ ■❢ |an| ≤ cn ❢♦r n ≥ N0, ✇❤❡r❡ N0 ✐s s♦♠❡ ✜①❡❞ ✐♥t❡❣❡r✱ ❛♥❞ ✐❢ cn ❝♦♥✈❡r❣❡s✱ t❤❡♥ an ❝♦♥✈❡r❣❡s✳ ✭❜✮ ■❢ an ≥ dn ≥ 0 ❢♦r n ≥ N0, ❛♥❞ ✐❢ dn ❞✐✈❡r❣❡s✱ t❤❡♥ an ❞✐✈❡r❣❡s✳

• ❡♦♠❡tr✐❝ s❡r✐❡s

◮ ■❢ 0 ≤ x < 1, t❤❡♥

∞

• n=0

xn = 1 1 − x. ■❢ x ≥ 1, t❤❡ s❡r✐❡s ❞✐✈❡r❣❡s✳

◮ Pr♦♦❢ ■❢ x = 1, ✇❡ ❤❛✈❡

sn =

n

• k=0

xk = 1 + x + x2 + x3 · · · + xn. ✭✷✸✮ ■❢ ✇❡ ♠✉❧t✐♣❧② ✭✷✸✮ ❜② x ✇❡ ❤❛✈❡ xsn = x + x2 + x4 · · · xn+1. ✭✷✹✮ ❆♣♣❧②✐♥❣ ✭✷✸✮−✭✷✹✮ ✇❡ ❤❛✈❡

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✻✹ ✴ ✽✶

◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ❙❡r✐❡s

sn − xsn = 1 − xn+1 sn(1 − x) = 1 − xn+1 sn = 1 − xn+1 1 − x . ❚❤❡ r❡s✉❧t ❢♦❧❧♦✇s ✐❢ ✇❡ ❧❡t n → ∞.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✻✺ ✴ ✽✶

◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ❚❤❡ ❘♦♦t ❛♥❞ ❘❛t✐♦ ❚❡sts

✸✳✼ ❚❤❡ ❘♦♦t ❛♥❞ ❘❛t✐♦ ❚❡sts

❚❤❡♦r❡♠ ✷✸

✭❘♦♦t ❚❡st✮ ●✐✈❡♥ an, ♣✉t α = limn→∞ sup

n

• |an|. ❚❤❡♥

✭❛✮ ■❢ α < 1, an ❝♦♥✈❡r❣❡s❀ ✭❜✮ ■❢ α > 1, an ❞✐✈❡r❣❡s❀ ✭❝✮ ■❢ α = 1, t❤❡ t❡st ❣✐✈❡s ♥♦ ✐♥❢♦r♠❛t✐♦♥✳

❚❤❡♦r❡♠ ✷✹

✭❘❛t✐♦ ❚❡st✮ ❚❤❡ s❡r✐❡s an ✭❛✮ ❝♦♥✈❡r❣❡s ✐❢ lim

n→∞ sup

• an+1

an

• < 1,

✭❜✮ ❞✐✈❡r❣❡s ✐❢

• an+1

an

• ≥ 1 ❢♦r n ≥ n0✱ ✇❤❡r❡ n0 ✐s s♦♠❡ ✜①❡❞

✐♥t❡❣❡r✳ ❚❤❡ r❛t✐♦ t❡st ✐s ❢r❡q✉❡♥t❧② ❡❛s✐❡r t♦ ❛♣♣❧② t❤❛♥ t❤❡ r♦♦t t❡st✳ ❍♦✇❡✈❡r✱ t❤❡ r♦♦t t❡st ❤❛s ✇✐❞❡r s❝♦♣❡✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✻✻ ✴ ✽✶

◆✉♠❡r✐❝❛❧ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s ❚❤❡ ❘♦♦t ❛♥❞ ❘❛t✐♦ ❚❡sts

❊①❡r❝✐s❡s ❈❤❛♣t❡r ✸

✭✶✮ ▲❡t s ∈ R. ❛♥❞ sn = 1 + [(−1)n/n]✳ {sn} ✐s ❜♦✉♥❞❡❞ ❛♥❞ ✐ts r❛♥❣❡ ✐s ✜♥✐t❡❄ ❲❤✐❝❤ ✈❛❧✉❡ {sn} ❝♦♥✈❡r❣❡s t♦❄ ✭✷✮ ❲r✐t❡ ❛ ❉❡✜♥✐t✐♦♥ ❢♦r −∞ ❡q✉✐✈❛❧❡♥t t♦ ❉❡✜♥✐t✐♦♥ ✹✸✳ ✭✸✮ ❆♣♣❧② t❤❡ r♦♦t ❛♥❞ r❛t✐♦ t❡sts ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡r✐❡s ✭❛✮

1 2 + 1 3 + 1 22 + 1 32 + 1 23 + 1 33 + 1 24 + 1 34 + · · · ,

✭❜✮

1 2 + 1 + 1 8 + 1 4 + 1 32 + 1 16 + 1 128 + 1 64 + · · · ,

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✻✼ ✴ ✽✶

❈♦♥t✐♥✉✐t② ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s

✹✳ ❈♦♥t✐♥✉✐t②

✹✳✶ ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s

❉❡✜♥✐t✐♦♥ ✹✻

▲❡t X ❛♥❞ Y ❜❡ ♠❡tr✐❝ s♣❛❝❡s✿ s✉♣♣♦s❡ E ⊂ X, f ♠❛♣s E ✐♥t♦ Y ✱ ❛♥❞ p ✐s ❛ ❧✐♠✐t ♣♦✐♥t ♦❢ E. ❲❡ ✇r✐t❡ f(x) → q ❛s x → p✱ ♦r lim

x→p f(x) = q

✭✷✺✮ ✐❢ t❤❡r❡ ✐s ❛ ♣♦✐♥t q ∈ Y ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt②✿ ❋♦r ❡✈❡r② ε > 0 t❤❡r❡ ❡①✐sts ❛ δ > 0 s✉❝❤ t❤❛t dY (f(x), q) < ε ✭✷✻✮ ❢♦r ❛❧❧ ♣♦✐♥ts x ∈ E ❢♦r ✇❤✐❝❤ 0 < dX(x, p) < δ. ✭✷✼✮ dX ❛♥❞ dY r❡❢❡r t♦ t❤❡ ❞✐st❛♥❝❡s ✐♥ X ❛♥❞ Y ✱ r❡s♣❡❝t✐✈❡❧②✳ ✱ ❜✉t ♥❡❡❞ ♥♦t ❜❡ ❛ ♣♦✐♥t ♦❢ ✳ ▼♦r❡♦✈❡r✱ ❡✈❡♥ ✐❢ ✱ ✇❡ ♠❛② ✈❡r② ✇❡❧❧ ❤❛✈❡

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✻✽ ✴ ✽✶

❈♦♥t✐♥✉✐t② ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s

❆❧t❡r♥❛t✐✈❡ st❛t❡♠❡♥t ❢♦r ❉❡✜♥✐t✐♦♥ ✹✻ ❜❛s❡❞ ♦♥ (ε, δ) ❧✐♠✐t ❞❡✜♥✐t✐♦♥ ❣✐✈❡♥ ❜② ❇❡r♥❛r❞ ❇♦❧③❛♥♦ ✐♥ ✶✽✶✼✳ ■ts ♠♦❞❡r♥ ✈❡rs✐♦♥ ✐s ❞✉❡ t♦ ❑❛r❧ ❲❡✐❡rstr❛ss ✷

❉❡✜♥✐t✐♦♥ ✹✼

❚❤❡ ❢✉♥❝t✐♦♥ f ❛♣♣r♦❛❝❤❡s t❤❡ ❧✐♠✐t L ♥❡❛r c ♠❡❛♥s✿ ❢♦r ❡✈❡r② ε t❤❡r❡ ✐s s♦♠❡ δ > 0 s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ x, ✐❢ 0 < |x − c| < δ, t❤❡♥ |f(x) − L| < ε. f ❛♣♣r♦❛❝❤❡s L ♥❡❛r c ❤❛s t❤❡ s❛♠❡ ♠❡❛♥✐♥❣ ❛s t❤❡ ❊q✉❛t✐♦♥ ✭✷✽✮ lim

x→c f(x) = L.

✭✷✽✮

✷❆❞❞❛♣t❡❞ ❢r♦♠ ❙♣✐✈❛❦✱ ▼✳ ✭✶✾✻✼✮ ❈❛❧❝✉❧✉s✳ ❇❡♥❥❛♠✐♥✿ ◆❡✇ ❨♦r❦✳ Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✻✾ ✴ ✽✶

❈♦♥t✐♥✉✐t② ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s

❋✐❣✉r❡ ✹✿ ❲❤❡♥❡✈❡r ❛ ♣♦✐♥t x ✐s ✇✐t❤✐♥ δ ♦❢ c✱ f(x) ✐s ✇✐t❤✐♥ ε ✉♥✐ts ♦❢ L✳ ❙♦✉r❝❡✿ ❲✐❦✐♣❡❞✐❛✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✼✵ ✴ ✽✶

❈♦♥t✐♥✉✐t② ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s

❚❤❡♦r❡♠ ✷✺

▲❡t X, Y, E, f✱ ❛♥❞ p ❜❡ ❛s ✐♥ ❉❡✜♥✐t✐♦♥ ✹✻✳ ❚❤❡♥ lim

x→p f(x) = q

✭✷✾✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ lim

n→∞ f(pn) = q

✭✸✵✮ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ {pn} ✐♥ E s✉❝❤ t❤❛t pn = p, lim

n→∞ pn = p.

✭✸✶✮

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✼✶ ✴ ✽✶

❈♦♥t✐♥✉✐t② ▲✐♠✐ts ♦❢ ❋✉♥❝t✐♦♥s

❚❤❡♦r❡♠ ✷✻

❙✉♣♣♦s❡ E ⊂ X✱ ❛ ♠❡tr✐❝ s♣❛❝❡✱ p ✐s ❛ ❧✐♠✐t ♣♦✐♥t ♦❢ E, f ❛♥❞ g ❛r❡ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥s ♦♥ E, ❛♥❞ lim

x→p f(x) = A,

lim

x→p g(x) = B.

❚❤❡♥ ✭❛✮ lim

x→p(f + g)(x) = A + B;

✭❜✮ lim

x→p(fg)(x) = AB;

✭❝✮ lim

x→p

f g

• (x) = A

B , ifB = 0.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✼✷ ✴ ✽✶

❈♦♥t✐♥✉✐t② ❈♦♥t✐♥✉♦✉s ❋✉♥❝t✐♦♥s

✹✳✷ ❈♦♥t✐♥✉♦✉s ❋✉♥❝t✐♦♥s

❉❡✜♥✐t✐♦♥ ✹✽

❙✉♣♣♦s❡ X ❛♥❞ Y ❛r❡ ♠❡tr✐❝ s♣❛❝❡s✱ E ⊂ X, p ∈ E, ❛♥❞ f ♠❛♣s E ✐♥t♦

• Y. ❚❤❡♥ f ✐s s❛✐❞ t♦ ❜❡ ❝♦♥t✐♥✉♦✉s ❛t p ✐❢ ❢♦r ❡✈❡r② ε > 0 t❤❡r❡ ❡①✐sts ❛

δ > 0 s✉❝❤ t❤❛t dY (f(x), f(p)) < ε ❢♦r ❛❧❧ ♣♦✐♥ts x ∈ E ❢♦r ✇❤✐❝❤ dX(x, p) < δ. ■❢ f ✐s ❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r② ♣♦✐♥t ♦❢ E, t❤❡♥ f ✐s s❛✐❞ t♦ ❜❡ ❝♦♥t✐♥✉♦✉s ♦♥ E. f ❤❛s t♦ ❜❡ ❞❡✜♥❡❞ ❛t t❤❡ ♣♦✐♥t p ✐♥ ♦r❞❡r t♦ ❜❡ ❝♦♥t✐♥✉♦✉s ❛t p✳ f ✐s ❝♦♥t✐♥♦✉s ❛t p ✐❢ ❛♥❞ ♦♥❧② ✐❢ limx→p f(x) = f(p).

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✼✸ ✴ ✽✶

❈♦♥t✐♥✉✐t② ❈♦♥t✐♥✉♦✉s ❋✉♥❝t✐♦♥s

❚❤❡♦r❡♠ ✷✼

❙✉♣♣♦s❡ X, Y, Z ❛r❡ ♠❡tr✐❝ s♣❛❝❡s✱ E ⊂ X, f ♠❛♣s E ✐♥t♦ Y ✱ g ♠❛♣s t❤❡ r❛♥❣❡ ♦❢ f✱ f(E)✱ ✐♥t♦ Z, ❛♥❞ h ✐s t❤❡ ♠❛♣♣✐♥❣ ♦❢ E ✐♥t♦ Z ❞❡✜♥❡❞ ❜② h(x) = g(f(x)) (x ∈ E). ■❢ f ✐s ❝♦♥t✐♥✉♦✉s ❛t ❛ ♣♦✐♥t p ∈ E ❛♥❞ ✐❢ g ✐s ❝♦♥t✐♥✉♦✉s ❛t t❤❡ ♣♦✐♥t f(p), t❤❡♥ h ✐s ❝♦♥t✐♥✉♦✉s ❛t p✳ ❚❤❡ ❢✉♥❝t✐♦♥ h = f ◦ g ✐s ❝❛❧❧❡❞ t❤❡ ❝♦♠♣♦s✐t❡ ♦❢ f ❛♥❞ g.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✼✹ ✴ ✽✶

❈♦♥t✐♥✉✐t② ❈♦♥t✐♥✉✐t② ❛♥❞ ❈♦♠♣❛❝t♥❡ss

✹✳✸ ❈♦♥t✐♥✉✐t② ❛♥❞ ❈♦♠♣❛❝t♥❡ss

❉❡✜♥✐t✐♦♥ ✹✾

❆ ♠❛♣♣✐♥❣ f ♦❢ ❛ s❡t E ✐♥t♦ Rk ✐s s❛✐❞ t♦ ❜❡ ❜♦✉♥❞❡❞ ✐❢ t❤❡r❡ ✐s ❛ r❡❛❧ ♥✉♠❜❡r M s✉❝❤ t❤❛t |f(x)| ≤ M ❢♦r ❛❧❧ x ∈ E.

❚❤❡♦r❡♠ ✷✽

❙✉♣♣♦s❡ f ✐s ❛ ❝♦♥t✐♥✉♦✉s ♠❛♣♣✐♥❣ ♦❢ ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ X ✐♥t♦ ❛ ♠❡tr✐❝ s♣❛❝❡ Y ✳ ❚❤❡♥ f(X) ✐s ❝♦♠♣❛❝t✳

❚❤❡♦r❡♠ ✷✾

❙✉♣♣♦s❡ f ✐s ❛ ❝♦♥t✐♥✉♦✉s r❡❛❧ ❢✉♥❝t✐♦♥ ♦♥ ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ X✱ ❛♥❞ M = sup

p∈X

f(p), m = inf

p∈X f(p).

✭✸✷✮ ❚❤❡♥ t❤❡r❡ ❡①✐st ♣♦✐♥ts p, q ∈ X s✉❝❤ t❤❛t f(p) = M ❛♥❞ f(q) = m.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✼✺ ✴ ✽✶

❈♦♥t✐♥✉✐t② ❈♦♥t✐♥✉✐t② ❛♥❞ ❈♦♠♣❛❝t♥❡ss

❚❤❡ ❝♦♥❝❧✉s✐♦♥ ♠❛② ❛❧s♦ ❜❡ st❛t❡❞ ❛s ❢♦❧❧♦✇s✿ ❚❤❡r❡ ❡①✐st ♣♦✐♥ts p ❛♥❞ q ✐♥ X s✉❝❤ t❤❛t f(q) ≤ f(x) ≤ f(p) ❢♦r ❛❧❧ x ∈ X; t❤❛t ✐s✱ f ❛tt❛✐♥s ✐ts ♠❛①✐♠✉♠ ✭❛t p✮ ❛♥❞ ✐ts ♠✐♥✐♠✉♠ ✭❛t q✮✳

❉❡✜♥✐t✐♦♥ ✺✵

▲❡t f ❜❡ ❛ ♠❛♣♣✐♥❣ ♦❢ ❛ ♠❡tr✐❝ s♣❛❝❡ X ✐♥t♦ ❛ ♠❡tr✐❝ s♣❛❝❡ Y ✳ ❲❡ s❛② t❤❛t f ✐s ✉♥✐❢♦r♠❧② ❝♦♥t✐♥✉♦✉s ♦♥ X ✐❢ ❢♦r ❡✈❡r② ε > 0 t❤❡r❡ ❡①✐sts δ > 0 s✉❝❤ t❤❛t dY (f(p), f(q)) < ε ✭✸✸✮ ❢♦r ❛❧❧ p ❛♥❞ q ✐♥ X ❢♦r ✇❤✐❝❤ dX(p, q) < δ.

❚❤❡♦r❡♠ ✸✵

▲❡t f ❜❡ ❛ ❝♦♥t✐♥✉♦✉s ♠❛♣♣✐♥❣ ♦❢ ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ X ✐♥t♦ ❛ ♠❡tr✐❝ s♣❛❝❡ Y. ❚❤❡♥ f ✐s ✉♥✐❢♦r♠❧② ❝♦♥t✐♥✉♦✉s ♦♥ X.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✼✻ ✴ ✽✶

❈♦♥t✐♥✉✐t② ❈♦♥t✐♥✉✐t② ❛♥❞ ❈♦♥♥❡❝t❡❞♥❡ss

✹✳✹ ❈♦♥t✐♥✉✐t② ❛♥❞ ❈♦♥♥❡❝t❡❞♥❡ss

❚❤❡♦r❡♠ ✸✶

■❢ f ✐s ❛ ❝♦♥t✐♥✉♦✉s ♠❛♣♣✐♥❣ ♦❢ ❛ ♠❡tr✐❝ s♣❛❝❡ X ✐♥t♦ ❛ ♠❡tr✐❝ s♣❛❝❡ Y, ❛♥❞ ✐❢ E ✐s ❛ ❝♦♥♥❡❝t❡❞ s✉❜s❡t ♦❢ X✱ t❤❡♥ f(E) ✐s ❝♦♥♥❡❝t❡❞✳

❚❤❡♦r❡♠ ✸✷

✭■♥t❡r♠❡❞✐❛t❡ ❱❛❛❧✉❡ ❚❤❡♦r❡♠✮ ▲❡t f ❜❡ ❛ ❝♦♥t✐♥✉♦✉s r❡❛❧ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ✐♥t❡r✈❛❧ [a, b]✳ ■❢ f(a) < f(b) ❛♥❞ ✐❢ c ✐s ❛ ♥✉♠❜❡r s✉❝❤ t❤❛t f(a) < c < f(b)✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♣♦✐♥t x ∈ (a, b) s✉❝❤ t❤❛t f(x) = c.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✼✼ ✴ ✽✶

❈♦♥t✐♥✉✐t② ❉✐s❝♦♥t✐♥✉✐t✐❡s

✹✳✺ ❉✐s❝♦♥t✐♥✉✐t✐❡s ■❢ x ✐s ❛ ♣♦✐♥t ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ❛t ✇❤✐❝❤ f ✐s ♥♦t ❝♦♥t✐♥✉♦✉s✱ ✇❡ s❛② t❤❛t f ✐s ❞✐s❝♦♥t✐♥✉♦✉s ❛t x✳

❉❡✜♥✐t✐♦♥ ✺✶

▲❡t f ❜❡ ❞❡✜♥❡❞ ♦♥ (a, b)✳ ❈♦♥s✐❞❡r ❛♥② ♣♦✐♥t x s✉❝❤ t❤❛t a ≤ x < b. ❲❡ ✇r✐t❡ f(x+) = q ✐❢ f(tn) → q ❛s n → ∞, ❢♦r ❛❧❧ s❡q✉❡♥❝❡s {tn} ✐♥ (x, b) s✉❝❤ t❤❛t tn → x. ❚♦ ♦❜t❛✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ f(x−), ❢♦r a < x ≤ b✱ ✇❡ r❡str✐❝t ♦✉rs❡❧✈❡s t♦ s❡q✉❡♥❝❡s {tn} ✐♥ (a, x). ■t ✐s ❝❧❡❛r t❤❛t ❛♥② ♣♦✐♥t x ♦❢ (a, b)✱ lim

t→x f(t) ❡①✐sts ✐❢ ❛♥❞ ♦♥❧② ✐❢

f(x+) = f(x−) = lim

t→x f(t).

❉❡✜♥✐t✐♦♥ ✺✷

▲❡t f ❜❡ ❞❡✜♥❡❞ ♦♥ (a, b). ■❢ f ✐s ❞✐s❝♦♥t✐♥✉♦✉s ❛t ❛ ♣♦✐♥t x ❛♥❞ ✐❢ f(x+) ❛♥❞ f(x−) ❡①✐st✱ t❤❡♥ f ✐s s❛✐❞ t♦ ❤❛✈❡ ❛ ❞✐s❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ✜rst ❦✐♥❞✳ ❖t❤❡r✇✐s❡✱ ✐t ✐s ♦❢ t❤❡ s❡❝♦♥❞ ❦✐♥❞✳

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✼✽ ✴ ✽✶

❈♦♥t✐♥✉✐t② ▼♦♥♦t♦♥✐❝ ❋✉♥❝t✐♦♥s

✹✳✻ ▼♦♥♦t♦♥✐❝ ❋✉♥❝t✐♦♥s

❉❡✜♥✐t✐♦♥ ✺✸

▲❡t f ❜❡ r❡❛❧ ♦♥ (a, b). ❚❤❡♥ f ✐s s❛✐❞ t♦ ❜❡ ♠♦♥♦t♦♥✐❝❛❧❧② ✐♥❝r❡❛s✐♥❣ ♦♥ (a, b) ✐❢ a < x < y < b ✐♠♣❧✐❡s f(x) ≤ f(y).

❚❤❡♦r❡♠ ✸✸

▲❡t f ❜❡ ♠♦♥♦t♦♥✐❝❛❧❧② ✐♥❝r❡❛s✐♥❣ ♦♥ (a, b)✳ ❚❤❡♥ f(x+) ❛♥❞ f(x−) ❡①✐st ❛t ❡✈❡r② ♣♦✐♥t ♦❢ x ♦❢ (a, b). ▼♦r❡ ♣r❡❝✐s❡❧② sup

a<t<x

f(t) = f(x−) ≤ f(x) ≤ f(x+) = inf

x<t<b f(t).

✭✸✹✮ ❋✉rt❤❡r♠♦r❡✱ ✐❢ a < x < y < b, t❤❡♥ f(x+) ≤ f(x−). ✭✸✺✮

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✼✾ ✴ ✽✶

❈♦♥t✐♥✉✐t② ■♥✜♥✐t❡ ▲✐♠✐ts ❛♥❞ ▲✐♠✐ts ❛t ■♥✜♥✐t②

✹✳✼ ■♥✜♥✐t❡ ▲✐♠✐ts ❛♥❞ ▲✐♠✐ts ❛t ■♥✜♥✐t② ❋♦r ❛♥② r❡❛❧ ♥✉♠❜❡r x, ✇❡ ❤❛✈❡ ❛❧r❡❛❞② ❞❡✜♥❡❞ ❛ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ x t♦ ❜❡ ❛♥② s❡❣♠❡♥t (x − δ, x + δ).

❉❡✜♥✐t✐♦♥ ✺✹

❋♦r ❛♥② r❡❛❧ c, t❤❡ s❡t ♦❢ r❡❛❧ ♥✉♠❜❡rs x s✉❝❤ t❤❛t x > c ✐s ❝❛❧❧❡❞ ❛ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ +∞ ❛♥❞ ✐s ✇r✐tt❡♥ (c, +∞). ❙✐♠✐❧❛r❧②✱ t❤❡ s❡t (−∞, c) ✐s ❛ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ −∞.

❉❡✜♥✐t✐♦♥ ✺✺

▲❡t f ❜❡ ❛ r❡❛❧ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ E. ❲❡ s❛② t❤❛t f(t) → A as t → x ✇❤❡r❡ A ❛♥❞ x ❛r❡ ✐♥ t❤❡ ❡①t❡♥❞❡❞ r❡❛❧ ♥✉♠❜❡r s②st❡♠✱ ✐❢ ❢♦r ❡✈❡r② ♥❡✐❣❤❜♦r❤♦♦❞ U ♦❢ A t❤❡r❡ ✐s ❛ ♥❡✐❣❤❜♦r❤♦♦❞ V ♦❢ x s✉❝❤ t❤❛t V ∩ E ✐s ♥♦t ❡♠♣t②✱ ❛♥❞ s✉❝❤ t❤❛t f(t) ∈ U ❢♦r ❛❧❧ t ∈ V ∩ E, t = x.

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✽✵ ✴ ✽✶

❈♦♥t✐♥✉✐t② ■♥✜♥✐t❡ ▲✐♠✐ts ❛♥❞ ▲✐♠✐ts ❛t ■♥✜♥✐t②

❚❤r❡❡ ✐♠♣♦rt❛♥t t❤❡♦r❡♠s✳

❚❤❡♦r❡♠ ✸✹

■❢ f ✐s ❝♦♥t✐♥✉♦✉s ♦♥ [a, b] ❛♥❞ f(a) < 0 < f(b)✱ t❤❡♥ t❤❡r❡ ✐s s♦♠❡ x ✐♥ [a, b] s✉❝❤ t❤❛t f(x) = 0.

❚❤❡♦r❡♠ ✸✺

■❢ f ✐s ❝♦♥t✐♥✉♦✉s ♦♥ [a, b]✱ t❤❡♥ f ✐s ❜♦✉♥❞❡❞ ❛❜♦✈❡ ♦♥ [a, b]✱ t❤❛t ✐s✱ t❤❡r❡ ✐s s♦♠❡ ♥✉♠❜❡r N s✉❝❤ t❤❛t f(x) ≤ N ❢♦r ❛❧❧ x ✐♥ [a, b].

❚❤❡♦r❡♠ ✸✻

■❢ f ✐s ❝♦♥t✐♥✉♦✉s ♦♥ [a, b]✱ t❤❡♥ t❤❡r❡ ✐s s♦♠❡ ♥✉♠❜❡r y ✐♥ [a, b] s✉❝❤ t❤❛t f(y) ≥ f(x) ❢♦r ❛❧❧ x ✐♥ [a, b].

Pr♦❢✳ ❊r✐✈❡❧t♦♥ ✭❯❋❙❏✮ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❆✉❣✉st ✷✵✶✺ ✽✶ ✴ ✽✶