characterization of convergence of sequences in metric spaces

Let $(M,d)$ be a metric space, and let $\mathbb{N}=P_1\bigcup \mathrm{\cdots }\bigcup P_k$ be a partition of the set of natural numbers such that $P_i$ is infinite for every $i$, that is, there is a bijection $f_i:\mathbb{N}\to P_i$. Then, given a sequence ${(x_n)}_{n\in \mathbb{N}}$, it converges to $x\in M$ if and only if the subsequence

$${(x_{f_i(n)})}_n$$

converges to $x$ for every $i=1,\mathrm{\cdots },k$.

Examples

If you have a sequence ${(x_n)}_n$ and a natural number $k$, and you know that it converges to $x$ for every corresponding subsequence over the classes of remainders modulo $k$, then it converges to $x$.