solutionshw1-555-2011

solutionshw1-555-2011 - Solutions homework 1(1 Prove that-1...

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Solutions 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). Proof: Let O n = ( - 1 , 1 - 1 n ). Then [ - 1; 1) ⊂ ∪ n O n , but [ - 1; 1) can’t be covered by N n =1 O n for any N , as 1 - 1 2 N / ∈ ∪ N n =1 O n . (2) What is an interior point? Prove that 1 4 is an interior point of (0; 2]. Proof: p is an interior point of E if there exists > 0 such that ( p - , p + ) E . Let = 1 4 . Then ( 1 4 - , 1 4 + ) = (0 , 1 2 ) (0; 2]. (3) Let a 1 = 6 and a n +1 = 6 + a n for n 1. a. Show that a n 3 for all n 1. For n = 1 we have a 1 = 6 9 = 3. Assume now that a n 3. Then a n +1 = 6 + a n 6 + 3 = 3. Hence by induction a n 3 for all n 1. b. Show that { a n } is an increasing sequence. For n = 1 we get a 2 = p 6 + 6 6 = a 1 . Assume now that a n +1 a n . Then a n +2 = 6 + a n +1 6 + a n = a n +1 . It follows by induction that { a n } is an increasing sequence.
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