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?