Subsequence of a Sequence in a Set

Let $X$ be a set, and let $(x_m)_{m\in\mathbb{N}}$ be a sequence in $X$, indexed by the natural numbers carrying the addition of that definition and the order $<$.

A sequence $(n_k)_{k\in\mathbb{N}}$ in $\mathbb{N}$ is strictly increasing if $n_k<n_{k+1}$ for every $k\in\mathbb{N}$.

A subsequence of $(x_m)_{m\in\mathbb{N}}$ is a sequence in $X$ of the form $(x_{n_k})_{k\in\mathbb{N}}$ for some strictly increasing sequence $(n_k)_{k\in\mathbb{N}}$ in $\mathbb{N}$.