4130prelim1sols

# 4130prelim1sols - MATH 4130 HONORS INTRODUCTION TO ANALYSIS I PRELIM 1 THURSDAY Please attempt all questions You have 70 minutes You may use any

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Unformatted text preview: MATH 4130 HONORS INTRODUCTION TO ANALYSIS I. PRELIM 1. THURSDAY MARCH 11 2010 Please attempt all questions. You have 70 minutes. You may use any theorems from the lecture notes, but please clearly state any theorems which you use. (1) (9 marks) Let X = (0 , 1) ∪ [3 , 4] ⊂ R . State whether the following statements about X are true or false and give a brief reason in each case. (a) sup( X ) = 4. True, since 4 is clearly an upper bound and any upper bound is ≥ 4 , by definition. (b) X can be written as a union of open sets. False. Any union of open sets is open, and X is not open. (c) | X | = | R | . True. | X | ≤ | R | because X ⊂ R . Also, in the lectures we have constructed an injection P ( N ) → (0 , 1) . Therefore, | X | ≥ |P ( N ) | = | R | . (2) (19 marks) Let { x n } be a sequence of real numbers. (a) (3 marks) State what it means for { x n } to converge to the limit L ∈ R ....
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## This note was uploaded on 09/22/2010 for the course MATH 413 at Cornell University (Engineering School).

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4130prelim1sols - MATH 4130 HONORS INTRODUCTION TO ANALYSIS I PRELIM 1 THURSDAY Please attempt all questions You have 70 minutes You may use any

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