$f_n(x_n)\to f(x) $ implies $f$ continuous – a question about the proof

My question refers to If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous. (the first answer)

It’s written there that the hypotheses imply $(f_n)$ converges to $f$ pointwise. From this, we can choose a subsequence $(f_{n_{k}})$ of $(f_n)$ such that

$\left|f_{n_k} (x_k) -f(x_k)\right|<\epsilon$ for every $k$

Could you explain to me why is that. I’ve already posted a comment there but I’m not sure if anyone will reply, because it was last active 3 weeks ago.

I would really appreciate all your help.

Thank you.

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