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section11and12notes - Lectures 11 and 12 Definition:...

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Unformatted text preview: Lectures 11 and 12 Definition: subsequence and subsequential limit. 1 Examples: identify the subsequential limits of the sequences (−1)n , cos n3π , n . Theorem: if (sn ) converges to s, then all subsequences (snk ) converge to s. Application: find the limit of sn = x1/n where 0 < x < 1 is fixed. Application: prove that sn = (−1)n does not converge. Proposition: if all sn > 0, and inf {sn } = 0, then there exists a subsequence of (sn ) which converges to 0. Theorem: every sequence (sn ) has a monotonic subsequence. Theorem: every sequence contains a monotonic subsequence which converges to lim sup sn (similarly for lim inf sn ). Bolzano-Weierstrass theorem: every bounded sequence has a convergent subsequence. Theorem: let S be the set of all subsequential limits of the sequence (sn ). Then S in non-empty, sup S = lim sup sn , inf S = lim inf sn , and (sn ) converges iff S is a one-element set consisting of lim sn . Theorem: if sn → s > 0, and (tn ) is bounded, then lim sup sn tn = s lim sup tn . 1 ...
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This note was uploaded on 10/16/2011 for the course MATH 34 taught by Professor Wiley during the Winter '11 term at UC Merced.

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