Let $(X,d)$ be a metric space, let $(x_n)_{n=1}^\infty$ be a sequence in $X$, and let $x\in X$. If $x_n\to x$ in $(X,d)$, then for every strictly increasing sequence $(n_k)_{k=1}^\infty$ of natural numbers, the subsequence $(x_{n_k})_{k=1}^\infty$ converges to $x$ in $(X,d)$.