( R , < ) is a complete ordered set. Density of the rationals. - PDF document

� 1 ילרגטנאוילמסטיניפנא�ובשח �הרזח 5/6/2010 Dedekind Cuts ( �דניקדדיכתח ) A Dedekind cut (D-cut) (aka lower-D-cut ) is a nonempty set A � Q so that (i) ( −∞ , a ) := { q ∈ Q : q < a } ⊂ A ∀ a ∈ A ; (ii) ∄ maximal element in A (i.e. ∄ LUB A ∈ A ). Let R := { lower-D-cuts } and order R by inclusion (i.e. for A , B ∈ R , A < B if A � B ), then � ( R , < ) is a complete ordered set. � Density of the rationals. Suppose that A , B ∈ R and that A < B , then ∃ q ∈ Q so that A < ( −∞ , q ) < B .

Decimal representation ( �תינורשעהגצה ) of lower-D-cuts Decimal representation is a map π : R → Z × D N where D := { digits } = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } defined by π ( A ) = ( N ; d 1 , d 2 , . . . ) where n � d k max { a ∈ A : 10 n a ∈ Z } = N + ∀ n ≥ 1 . 10 k k =1 π : R → Z × { a ∈ D N : # { k ≥ 1 : a k ≥ 1 } = ∞} is a set • correspondence (bijection). � Cantor’s Theorem A non trivial interval in R is uncountably infinite. Addition in R Let A , B ∈ R be cuts. Define A + B := { a + b : a ∈ A , b ∈ B } , then , • A + B is a cut; neutral element for addition: 0 ∗ := ( −∞ , 0); • negative of D-cut A : − A := {− b : b ∈ A ↑ } which is is also a • cut, where; � A = ( −∞ , a ) , a ∈ Q , ( a , ∞ ) A ↑ := Q \ A A irrational ; A + ( − A ) = 0 ∗ . •

Positivity and order in R A D-cut is positive if A > 0 ∗ ( i.e. A � ( −∞ , 0) or 0 ∈ A ). Let R pos := { positive D-cuts } , then , ∀ A ∈ R , either A = 0 ∗ , A ∈ R pos or − A ∈ R pos ; • if A , B ∈ R pos , then A + B ∈ R pos ; • let A , B ∈ R , then A < B iff B + ( − A ) ∈ R pos . • Absolute value The absolute value of the D-cut A is | A | ∈ R pos defined by  0 A = 0 ,   A ∈ R pos , | A | = A   − A ∈ R pos . − A In particular (!) | ( −∞ , a ) | = ( −∞ , | a | ).

Multiplication We first define multiplication on R pos . Define the positive part of the positive D-cut A by A + := A ∩ (0 , ∞ ), then A = ( −∞ , 0] ∪ A + . For A , B ∈ R pos , define AB := ( −∞ , 0] ∪ { xy : x ∈ A + , y ∈ B + } and for A , B ∈ R define  0 A = 0 or B = 0 ,   A , B ∈ R pos or − A , − B ∈ R pos A · B := | A || B |   −| A || B | else . � Dedekind’s theorem ( R , + , · ) is a complete ordered field with respect to R pos . Archimedean ordered fields The ordered field ( F , + , · ) is called archimedean if ∀ x > 0 , ∃ n ∈ N such that nx := x + · · · + x > 1; � �� � n times there are no ‘‘ infinitesimals ") . (i.e. � A complete, ordered field ( F , + , · ) is archimedean.

Non-archimedean ordered field Let ( F , + , · ) be the field of rational functions } on R equipped with regular addition and multiplication of functions. Define the “ positive elements ” of F by F pos := { R = P Q ∈ F : ∃ t > 0 so that Q ( x ) � = 0 & R ( x ) > 0 ∀ x ∈ (0 , t ) } . � F pos is an ordering for ( F , + , · ); � F is not archimedean with respect to F pos . The complex numbers ( ��יבכורמה�ירפסמה ) Define the complex numbers by √ √ C = R ( − 1 y : x , y ∈ R } , − 1) := { x + with addition and multiplication to satisfy the normal laws of arithmetic. � ( C , + · ) is a field; � there is no ordering for ( C , + · )

General existence of real roots n √ a ∈ R pos such that( n √ a ) n = a . ∀ a ∈ R pos n ∈ N , ∃ ! Proof n √ a := LUB A where A := { x ∈ R pos : x n < a } . Limit of a sequence Suppose b n ∈ R ( n ∈ N ). We say that b n tends to ( ��ל�אוש ) B ∈ R as n → ∞ written b n → B ∈ R as n → ∞ ; or b n − → n →∞ B if ∀ ǫ > 0 , ∃ n ǫ such that | b n − B | < ǫ ∀ n ≥ n ǫ . Example If a n ≤ a n +1 and { a n : n ≥ 1 } is bounded, then a n − n →∞ LUB { a n : n ≥ 1 } . →

Conditions for convergence ¶ Comparison Suppose that a n ≥ 0 , a n → 0 as n → ∞ and that M > 0 , b n ∈ R , | b n | ≤ Ma n ∀ n ≥ 1, then b n → 0 as n → ∞ . ¶ Absolute value proposition a n → L as n → ∞ iff | a n − L | → 0 as n → ∞ and in this case | a n | → | L | . ¶ Sandwich principle Suppose that a n ≤ x n ≤ b n ∀ n ≥ 1 and that a n → L , b n → L as n → ∞ , then x n − n →∞ L . → Divergence to ∞ We say that the sequence ( x 1 , x 2 , . . . ) diverges ( �תרדבתמ ) to ∞ (as n → ∞ ) if for each M > 0 , ∃ N M such that x n > M ∀ n ≥ N M (and write this x n → ∞ ). � Let ( x 1 , x 2 , . . . ) be an increasing sequence, then either ( x 1 , x 2 , . . . ) is convergent, or x n → ∞ . Examples n n � � 1 1 k − n →∞ ∞ , → 2 k − n →∞ 2 . → k =1 k =0

Arithmetic of limits Suppose that a n → a and b n → b as n → ∞ , then a n + b n → a + b as n → ∞ ; (1) a n b n → ab as n → ∞ ; (2) and in case b � = 0: a n → a b as n → ∞ ; (3) b n Accumulation points Let E ⊂ R . A point x ∈ R is called an accumulation point of E if ∀ ǫ > 0 , # E ∩ ( x − ǫ, x + ǫ ) = ∞ . � The following are equivalent for E ⊂ R and x ∈ R : (i) x is an accumulation point of E ; (ii) ∀ ǫ > 0 , ∃ y ∈ E ∩ ( x − ǫ, x + ǫ ) , y � = x ; (iii) ∃ ( z 1 , z 2 , . . . ) ∈ E N such that z k � = z ℓ ∀ k � = ℓ & z n − n →∞ x . → � Bolzano-Weierstrass theorem (accumulation points) If E is an infinite, bounded set, then E ′ � = ∅ . The proof of the Bolzano-Weierstrass theorem uses: � Cantor’s Lemma (or the Chinese box theorem) A nested sequence of non-empty, closed intervals in R has a non-empty intersection.

Proof of the Bolzano-Weierstrass theorem Suppose that E ⊂ I a closed, finite interval. For I = [ a , b ], write I − := [ a , a + b 2 ] and I + := [ a + b 2 , b ]; I = I − ∪ I + whence ∃ I 1 = I ± with # E ∩ I 1 = ∞ ; Similarly ∃ I 2 = I ± 1 with # E ∩ I 2 = ∞ ; continuing, get closed intervals I n ⊃ I n +1 so that I n +1 = I ± • n and # E ∩ I n = ∞ ∀ n ≥ 1. n →∞ 0, by Cantor’s lemma, � ∞ Since | I n | = | I | 2 n − → n =1 I n = { Z } for some Z ∈ R . Z ∈ E ′ . � � Limit points Let E ⊂ R . A point x ∈ R is called a limit point ( �לובגתדוקנ ) of E if ∃ y n ∈ E ( n ≥ 1) such that y n → x . ( �רוגס ) of E : Closure { limit points of E } =: E . � E = E ∪ E ′ A set is closed if E = E . � A closed subset of R which is bounded above (below) has a maximal (minimal) element.

Subsequences An integer subsequence ( ��ימלשלשהרדיס�תת ) is an infinite subset K ⊂ N , K = { n 1 , n 2 , . . . } arranged in increasing order n 1 < n 2 < · · · → ∞ . A subsequence of the sequence { a 1 , a 2 , . . . } is a sequence of form { a n 1 , a n 2 , . . . } where n k → ∞ is an integer subsequence. � Bolzano-Weierstrass Theorem (convergent subsequences) Every bounded sequence has a convergent subsequence. Partial limits of a sequence The partial limit set ( ��ייקלחהתולובגתצובק ) PL ( a 1 , a 2 , . . . ) of the bounded sequence ( a 1 , a 2 , . . . ) is PL ( a 1 , a 2 , . . . ) := { a ∈ R : ∃ n k → ∞ , a n k − k →∞ a } � = ∅ . → � For ( a 1 , a 2 , . . . ) a bounded sequence, # PL ( a 1 , a 2 , . . . ) = 1 ⇐ ⇒ ∃ lim n →∞ a n . � Let ( a 1 , a 2 , . . . ) be a bounded sequence, then PL ( a 1 , a 2 , . . . ) is closed and bounded.

Upper and lower limits The upper limit ( ��וילעלובג ) of the bounded sequence ( a 1 , a 2 , . . . ) is n →∞ a n := max PL ( a 1 , a 2 , . . . ) lim and the lower limit ( ��ותחתלובג ) of the sequence ( a 1 , a 2 , . . . ) is lim a n := min PL ( a 1 , a 2 , . . . ) . n →∞ � ( a 1 , a 2 , . . . ) converges iff lim n →∞ a n = lim n →∞ a n . � Let a = ( a 1 , a 2 , . . . ) be a bounded sequence, then (i) ∀ α < lim n →∞ a n , ∃ N α such that a n > α ∀ n > N α ; (ii) ∀ β > lim n →∞ a n , K ≥ 1 , ∃ N > K such that a N < β ; (i) ∀ ω > lim n →∞ a n , ∃ N ω such that a n < ω ∀ n > N ω ; (ii) ∀ ξ < lim n →∞ a n , K ≥ 1 , ∃ N > K such that a N > ξ ; Cauchy sequences Or how to prove a sequence converges without knowing the limit. A sequence ( a 1 , a 2 , . . . ) is called a Cauchy sequence if ∀ ǫ > 0 , ∃ N ǫ ≥ 1 such that | a n − a n ′ | < ǫ ∀ n , n ′ ≥ N ǫ . � Cauchy’s Theorem A sequence converges ⇐ ⇒ it is a Cauchy sequence.