Unformatted text preview: ASSIGNMENT 2 Â· SOLUTIONS MAT 472 Â· FALL 2011 Problem 1 (Exercise 1.3.4) . Suppose A and B are nonempty subsets of R which are both bounded above, and such that B âŠ† A . Prove that sup B â‰¤ sup A . Proof. Since a â‰¤ sup A for all a âˆˆ A and B âŠ† A , we certainly have b â‰¤ sup A for all b âˆˆ B . Thus sup A is an upper bound for B . By definition, then, sup B â‰¤ sup A . Problem 2 (Exercise 1.3.6) . Compute (without proofs) the suprema and infima: (a) A = { n âˆˆ N  n 2 < 10 } (b) B = { n/ ( m + n )  m,n âˆˆ N } (c) C = { n/ (2 n + 1)  n âˆˆ N } (d) D = { n/m  m,n âˆˆ N and m + n â‰¤ 10 } . Solution. (a) Since A = { 1 , 2 , 3 } is finite, sup A = max A = 3 and inf A = min A = 1. (b) Clearly 0 â‰¤ b â‰¤ 1 for all b âˆˆ B . But elements of B get arbitrarily close to 0 (fix n and take m large), and (other) elements of B get arbitrarily close to 1 (fix m and take n large). Thus inf B = 0 and sup B = 1....
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 Spring '06
 Spielberg
 Sets, Supremum, infimum, S. Kaliszewski, School of Mathematical and Statistical Sciences

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