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4130prelim1 - L(c(8 marks Let x 1 ∈ R and define a...

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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. (b) X can be written as a union of open sets. (c) | X | = | 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 . (b) (8 marks) Let k N and define a sequence { y n } by y n = x n + k , n 1. Suppose { x n } converges to L . Show that { y n } also converges to
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Unformatted text preview: L . (c) (8 marks) Let x 1 ∈ R and define a sequence of real numbers { x n } by x n +1 = x 2 n + x n + 1 , n ≥ 1 . Show that the sequence { x n } does not converge. (3) (22 marks) Let A ⊂ R . (a) (3 marks) Explain what it means to say that x ∈ R is a cluster point (a.k.a. limit-point; accumulation point) of A . (b) (3 marks) Explain what it means to say that the set A is bounded . (c) Now let S = { x ∈ R : x is a cluster point of A } . (i) (8 marks) Show that S is a closed set. (ii) (8 marks) Suppose A is bounded. Show that S is a compact set. [END OF PAPER]...
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