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I came across the following problem about cluster points:

Prove the following: $K$ is a cluster point $\Longleftrightarrow$ $K$ is the limit of some subsequence $\{a_{n_i}\}$.

This is my attempt:

Proof. $(\Leftarrow)$: Suppose $K$ is the limit of some subsequence $\{a_{n_i}\}$. Then for each $\epsilon >0$ there exists a $N$ such that for all $i >N$ we have $|a_{n_i}-K| < \epsilon$. This happens for infinitely many $i$. Hence $K$ is a cluster point. ($\Rightarrow)$: Suppose $K$ is a cluster point. Then given $\epsilon >0$, $|a_n-K| < \epsilon$ for infinitely many $n$. Pick $n_1$ such that $|a_{n_1}-K| < 1$. Pick $n_2$ such that $|a_{n_2}-K| < \frac{1}{2}$ where $n_2>n_1$. Keep doing this (letting $\epsilon$ get smaller and smaller). If follows that $K$ is the limit of the subsequence we constructed. QED

Is this correct?

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  • $\begingroup$ "keep doing this" and "$K$ is the limit of the subsequence we constructed" is a bit vague. Try to make this precise. $\endgroup$ Jul 3, 2011 at 19:44
  • $\begingroup$ Ideally you should indicate what definition of "cluster point" you're using (when I first read this I thought you were proving a tautology), say what $K$ is, indicate where the terms of the subsequence are taken from (presumably from some unnamed set), what the context is (real line? metric space?). However, these are clean-up issues to deal with once you get the essential mathematical ideas down, not things you should be overly concerned with in an initial write-up. $\endgroup$ Aug 2, 2011 at 20:47

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The proof is correct and well written. It might be helpful for a reader who's learning the subject for the first time to explain at the end of your proof of ($\Rightarrow$), why the subsequence $a_{n_i}$ converges to $K$. Here we're only choosing $\epsilon$ to be $1$, $1/2$, etc. What happens when you're given an arbitrary $\epsilon$?

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