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Unformatted text preview: 1. (10 pts) Suppose that { x n } , { y n } , and { z n } are real sequences, and that for all positive integers n , we have x n ≤ y n ≤ z n . If both { x n } and { z n } converge and have the same limit, L , prove that { y n } also converges, and its limit is L . Solution: Let ε > 0. Since x n → L , there exists N ∈ N such that if n > N then  x n L  < ε . Similarly, since z n → L , there exists M ∈ N such that if n > M then  z n L  < ε . Now suppose that n > max( N,M ). Then ε < x n L  ≤ x n L ≤ y n L ≤ z n L ≤  z n L  < ε, ie  y n L  < ε. 2. (10 pts) Prove directly from the definition of compactness that R is not compact. Solution: Consider the collection of open sets O = { ( n 1 ,n + 1) } n ∈ N . Clearly O forms an open cover of R . Notice that any n ∈ N is in exactly one set ( m 1 ,m + 1), namely when m = n . It follows that we cannot remove any set in O and still be left with a cover of R . Hence O has no proper subcovers; in particular, it has no finite subcovers....
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This note was uploaded on 10/18/2011 for the course MATH S4061X taught by Professor Peters during the Spring '11 term at Columbia.
 Spring '11
 Peters
 Integers

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