**Lemma**: If $a_{n} \to l$, then any [[Real Subsequence|subsequence]] [[Convergence|converges]] to $l$. **Proof**: Let $(a_{n_{j}})_{j}$ be a subsequence of $(a_{n})$. Take $\epsilon>0$. Since $a_{n} \to l$, $\exists N$ such that $|a_{n}-l|<\epsilon, \quad \forall n\geq N$. Now since $n_{j} \to \infty$, $\exists J$ such that $n_{j}\geq N \quad \forall j \geq J$. It follows:$|a_{n_{j}}-l|<\epsilon \quad \forall j\geq J$So $a_{n_{j}} \to l$ as $j \to \infty$.