solutionshw5-201141

# solutionshw5-201141 - Solutions for HW 5 Problem 66 3...

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Unformatted text preview: Solutions for HW 5 Problem 66: 3. Define: f n ( x ) = 1 for 1 n < x ≤ 1 2 nx- 1 for 2 n < x ≤ 1 n for 0 ≤ x ≤ 2 n . Then f n ∈ B ( φ, 1), but ( f n ) has no uniformly convergent subsequence, as ρ ( f n ,f m ) = 1 for all n 6 = m , where ρ denotes the uniform metric. Problem 66: 5. (1) Let { F α } be a collection of compact sets. The F α is closed for each α , so F = ∩ α F α is closed. Now F is a closed subset of any of the compact sets F α and is thus also compact. (2) Assume F k ( k = 1 , ··· ,n ) is compact. Let F = ∪ n k =1 F k and let C be an open covering of F . Then C is an open covering of each of the F k , so has a finite subcovering C k . Then C 1 ∪···C n is a finite subcovering of F . (Alternate proof using sequential compactness: let ( x n ) be a sequence in F . Then there exists a k such that F k contains infinitely many terms of ( x n ), i.e. there is a subsequence ( x n k ) which is contained in F k . This subsequence has another subsequence (...
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