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Unformatted text preview: creasing sequences). If 6 = S R and b = sup S, then there is a sequence x n S such that x n b . (You should be able to state and prove the corresponding result for the inmum). If x n is a convergent sequence, then it must be Cauchy. If x n is a Cauchy sequence, then it must be bounded. If x n L ,. then for any subsequence x n k , x n k L. Every sequence has a monotonic subsequence. If x n is a Cauchy sequence and a subsequence x n k L, then x n L. If x n is a Cauchy sequence, then it must converge. If x n is a bounded sequence, then it has a convergent subsequence. [This is often called the Balzano-Weierstrass Theorem ] 1...
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- Winter '10