A Subsequence of a Subsequence is a Subsequence

Let $\mathbb{N}$ be the set of natural numbers with the order $<$ of that definition, let $X$ be a set, let $(x_m)_{m\in\mathbb{N}}$ be a sequence in $X$, and let $(n_k)_{k\in\mathbb{N}}$ and $(k_j)_{j\in\mathbb{N}}$ be sequences in $\mathbb{N}$ that are strictly increasing. Then the following hold.

1. (Order preservation) For all $a,b\in\mathbb{N}$ with $a<b$ one has $n_a<n_b$.

2. (Composite index sequence) The sequence $(n_{k_j})_{j\in\mathbb{N}}$ in $\mathbb{N}$ is strictly increasing.

3. (Composite subsequence) The sequence $(x_{n_{k_j}})_{j\in\mathbb{N}}$ is a subsequence of $(x_m)_{m\in\mathbb{N}}$, and it is the subsequence of $(x_{n_k})_{k\in\mathbb{N}}$ determined by the strictly increasing sequence $(k_j)_{j\in\mathbb{N}}$.