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Math Help - sequence, convergence

  1. #1
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    sequence, convergence

    Let (x_n) be a sequence of real numbers.

    (a) Prove that (x_n) converges to a number A iff every subsequence of (x_n) has a subsequence which converges to A.
    (b) Does (x_n) converge if every subsequence of (x_n) has a subsequence which converges?
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  2. #2
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    I figured out part b, but part a is confusing me.
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  3. #3
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    Quote Originally Posted by poincare4223 View Post
    (a) Prove that (x_n) converges to a number A iff every subsequence of (x_n) has a subsequence which converges to A.
    If \{x_n\} is convergent then all subsequences converge to the same limit. Thus, the forward direction follows.

    To prove the backwards direction assume that \{x_n\} [u]did not[/b] converge to A. Then it means there is some \epsilon > 0 so that for any N>0 we have |x_n - A| \geq \epsilon for some n>N. Thus, we can form a subsequence, \{x_{n_k}\} with |x_{n_k} - A|\geq \epsilon. But this subsequence cannot possible have a subsequence converging to A . Contradiction.
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