Every lingering sequence has a convergent subsequence/Proof

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TODO: Write proof

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Proof outline:

1. Take [ilmath]k_1[/ilmath] to be the index of any point of the sequence in [ilmath]B_1(x)[/ilmath]
2. Take [ilmath]k_2[/ilmath] to be any index AFTER [ilmath]k_1[/ilmath] of the sequence in the ball [ilmath]B_\frac{1}{2}(x)[/ilmath]
3. ...
4. Show the sequence [ilmath](x_{k_n})_{n=1}^\infty[/ilmath] converges to [ilmath]x[/ilmath]

We have exhibited a convergent subsequence, we're done.