Every lingering sequence has a convergent subsequence/Proof

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TODO: Write proof

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Proof outline:

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

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