Every Subsequence of a Convergent Sequence converges to the same Limit

Last updated Nov 1, 2022

Let $(a_{n}) \subset X$ be a sequence in Metric Space $(X, d)$. If $a_{n} \to a \in X$ every Subsequence of $(a_{n})$ converges to $a$.

Suppose $a_{n} \to a$. Let $(a_{n_{k}})$ be a Subsequence of $(a_{n})$. Since $a_{n} \to a$, we know $\forall \epsilon > 0$, $\exists \mathbb{N}\in N$ s.t. $\forall n \geq N$ we have that $$d(a_{n}, a) < \epsilon$$ Let $K = \min{k \in \mathbb{N} : n_{k} \geq N}$. Then we have that $$d(a_{n_{k}}, a) < \epsilon$$ for all $k \geq K$ since $n_{k} \geq N$ by our choice of $K$. Thus $a_{n_{k}} \to a$ as well. $\blacksquare$

• Natural Numbers
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1. Pugh - Real Mathematical Analysis - Ch 2, pg unknown