How would I prove this statement? Every subsequence of $x_n$ has a further subsequence which converges to $x$.Then the sequence $x_n$ converges to $x$. the post here explains one way, but how would I prove the other direction?
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1The other direction is trivial: if $x_n \to x$, then every subsequence of $x_n$ converges to $x$, hence every subsequence has a subsequence (namely itself) which converges to $x$. – Mar 06 '17 at 06:31