How can I prove that a sequence in a metric space converges to an element iff every sub-sequence has a sub-sub-sequence convergent to that element. Thank for the help!!
2026-04-07 06:09:43.1775542183
exercise on convergent sequences
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Hint: If $x_n$ does not converge to $L,$ then for some $\epsilon>0,$ there is no $N$ such that $n\ge N$ implies $d(x_n,L) <\epsilon.$ Argue that this implies there is a subsequence $x_{n_k}$ such that $d(x_{n_k},L) \ge\epsilon$ for all $k.$