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hw1-555-2011 - Homework 1(1 Prove that-1 1 is not compact...

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Unformatted text preview: Homework 1. (1) Prove that [-1, 1) is not compact by using the definition of a compact set (to get credit for the problem, use the definition and not any theorems about compact sets). (2) What is an interior point? Prove that 1 4 is an interior point of (0, 2]. (3) Let a1 = 6 and an+1 = 6 + an for n 1. a. Show that an 3 for all n 1. b. Show that {an } is an increasing sequence. c. Explain why {an } converges. d. Determine the value of limn an . (4) Complete the table below indicating Int(E) (the interior of E), and whether E is open, closed, both, or neither. An answer of "open" or "closed" in the next to last column will mean that you think E is "open and not closed" or "closed and not open" respectively. You do not need to show work on this problem. E Int(E) Open? or Closed? Compact (-1, 1] (0, ) Q R [2, ) (5) Suppose p is in the closure of two sets A and B. a. Must p be in the closure of A B? Justify your answer. b. Must p be in the closure of A B? Justify your answer. 1 ...
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