Definition of subsequence used in defining accumulation points

A subsequence of a sequence $(a_n)_{n\in\Bbb N}$ is a sequence $(a_{n_k})_{k\in\Bbb N}$ such that $(n_k)_{k\in\Bbb N}$ is a strictly increasing sequence of natural numbers. So, for instance, $(a_{2n})_{n\in\Bbb N}$ and $(a_{n^2})_{n\in\Bbb N}$ are subsequence of $(a_n)_{n\in\Bbb N}$. In the first case, I took $n_k=2k$ and, in the second case, I took $n_k=k^2$.

A formal way to define a subsequence is the following.

First let us recall what is the formal definition of a sequence. A sequence in a set $X$ is a function $f:\mathbb N\to X$. One may write $x_n$ to mean $f(n)$, and $f$ now may be denoted as $(x_n)_{n\in \mathbb N}$.

A strictly monotonically increasing function from $\mathbb N$ to $\mathbb N$ is a map $\phi:\mathbb N\to \mathbb N$ such that $\phi(i)<\phi(j)$ whenever $i<j$.

Now, if $f$ is a sequence in $X$, a subsequence of $f$ is any function of the form $f\circ \phi$, where $\phi:\mathbb N\to \mathbb N$ is a strictly mononotonically increasing function. If $(x_n)$ denotes the sequence $f$, and $\phi(i) = n_i$, then we may denote the seqeunce $f\circ \phi$ as $(x_{n_i})_{i\in \mathbb N}$