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Presentation Presentation Kim Do kyoung POSTECH May 1, 2008 Presentation Introduction Math Analysis Suppose Then n f n ( x ) = 0 lim x [0 , 1] 1 lim f n ( x ) dx = 0 ? where f n : [0 , 1] R is n 0

  1. Presentation Presentation Kim Do kyoung POSTECH May 1, 2008

  2. Presentation Introduction Math Analysis Suppose Then n →∞ f n ( x ) = 0 lim ∀ x ∈ [0 , 1] � 1 ⇒ lim f n ( x ) dx = 0 ? where f n : [0 , 1] → R is n →∞ 0 continuous

  3. Presentation Introduction Math Analysis Suppose Then n →∞ f n ( x ) = 0 lim ∀ x ∈ [0 , 1] � 1 ⇒ lim f n ( x ) dx = 0 ? where f n : [0 , 1] → R is n →∞ 0 continuous The answer is

  4. Presentation Introduction Math Analysis Suppose Then n →∞ f n ( x ) = 0 lim ∀ x ∈ [0 , 1] � 1 ⇒ lim f n ( x ) dx = 0 ? where f n : [0 , 1] → R is n →∞ 0 continuous The answer is, of course,

  5. Presentation Introduction Math Analysis Suppose Then n →∞ f n ( x ) = 0 lim ∀ x ∈ [0 , 1] � 1 ⇒ lim f n ( x ) dx = 0 ? where f n : [0 , 1] → R is n →∞ 0 continuous The answer is, of course, No.

  6. Presentation Introduction Math Counterexample ( n + 1) 2 ( x − 1 1 2  n +1 ) , n +1 ≤ x ≤ n +1  3 2 3 ( n + 1) 2 ( − x + Let f n ( x ) = n +1 ) , n +1 ≤ x ≤ n +1 0 , otherwise  The following figure is when n = 4 .

  7. Presentation Introduction American History The Statue of Liberty The tablet contains the text ”JULY IV MDCCLXXVI” (July 4, 1776) commemorating the date of the United States Declaration of Independence.

  8. Presentation Introduction American History The Statue of Liberty

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