Subsequence

Let $(a_{n})_{n\in \N}$ be a sequence in a set $X$. A subsequence of $(a_{n})$ is a sequence of the form
$$(a_{n_{k}})_{k\in \N}$$where $(n_{k})$ is a strictly increasing sequence of natural numbers.

Sometimes, one also uses the notation $(a_{\varphi(n)})_n$ where $\varphi(n)\colon\N\to\N$ is a strictly increasing function called "extractor".