Subsequences of convergent sequences converge to the same limit

Proposition (Subsequence inherits the limit). Let $(X,d)$ be a metric space , and let $(x_n)$ be a sequence in $X$ that converges to $a\in X$. If $(x_{n_k})$ is a subsequence of $(x_n)$, then $x_{n_k}\to a$.

Context. This is the basic “stability” property of limits under passing to subsequences. It is used constantly in compactness arguments and in extracting limits from sequences.

Proof sketch. Fix $\varepsilon>0$. Since $x_n\to a$, there exists $N$ such that $d(x_n,a)<\varepsilon$ for all $n\ge N$. Because $(n_k)$ is increasing and unbounded, there exists $K$ such that $n_k\ge N$ for all $k\ge K$. Then $d(x_{n_k},a)<\varepsilon$ for all $k\ge K$, so $x_{n_k}\to a$.

Example. In $\mathbb{R}$, if $x_n=1/n\to 0$, then any subsequence $x_{n_k}=1/n_k$ also converges to $0$.