A Subsequence of a Convergent Sequence Has the Same Limit

Let $(X,d)$ be a metric space, let $(x_m)_{m\in\mathbb{N}}$ be a sequence in $X$, and let $x\in X$ be such that $(x_m)_{m\in\mathbb{N}}$ converges to $x$ in $(X,d)$. Let $(n_k)_{k\in\mathbb{N}}$ be a strictly increasing sequence in $\mathbb{N}$, so that $(x_{n_k})_{k\in\mathbb{N}}$ is a subsequence of $(x_m)_{m\in\mathbb{N}}$.

Then $(x_{n_k})_{k\in\mathbb{N}}$ converges to $x$ in $(X,d)$.