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Let {${a_n}$} be sequence of real number.

I understand the the fact that

(ⅰ) {${a_n}$} converges to $a$ when $n→∞$

(ⅱ)For arbitrary strictly increasing sequence of natural numbers ${n(i)}$, there exists strictly increasing sequence of natural numbers ${n(j)}$, such that {${a_n(i(j))}$} converges to $a$ when $j→∞$.

My question : What kind of benefit from characterizing congvergence like (ⅱ)? Also, I have never seen this characterization in any reference. If you could find any refference about this, I would be grateful if you could tell me.

BrauerManinobstruction
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    This is usually stated as an exercise I guess.. I remember using this to prove that every injective continuous function $f : E\to \mathbb R$ from a compact set in $E\subset \mathbb R$ has a continuous inverse. – Arctic Char Jul 12 '21 at 09:28
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    Some reference in MSE: [here](https://math.stackexchange.com/questions/397978/every-subsequence-of-x-n-has-a-further-subsequence-which-converges-to-x-the) – Arctic Char Jul 12 '21 at 09:29

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