Given a sequence $\{x_{n}\}_{{n\in\mathbb{N}}}$, any infinite subset of the sequence forms a subsequence. We formalize this as follows:

Equivalently, $\{y_{n}\}_{{n\in\mathbb{N}}}$ is a subsequence of $\{x_{n}\}_{{n\in\mathbb{N}}}$ if

1. 1.

   $\{y_{n}\}_{{n\in\mathbb{N}}}$ is a sequence of elements of $X$, and

2. 2.

   there is a strictly increasing function $a:\mathbb{N}\to\mathbb{N}$ such that

   $$y_{n}=x_{{a(n)}}\quad\text{ for all }n\in\mathbb{N}.$$