Sequence

A sequence is a function is any function $$ f$$ such that $$ \text{dom}\left(f\right)=I$$ where $$ I$$ either finite or countable.

With this in place, we can consider the $$ n$$-th element by $$ f\left(n\right)$$. Since we can think of the sequence as being enumerated, the notation $$ {a}_{n}$$ is employed for $$ f\left(n\right)$$ where $$ a$$ is usually any choice of lower case alphabet characters like $$ a,b,c,d,\dots $$. That said, we can notate the entire sequence by $$ \left({a}_{n}\right)$$

Subsequence

Given a sequence $$ \left({a}_{n}\right):{\mathbb{N}}_{1}\to \mathbb{R}$$ then given any strictly increasing map $$ \sigma :{\mathbb{N}}_{1}\to {\mathbb{N}}_{1}$$ we say that the sequence $$ {x}_{n}:=\left({a}_{\sigma \left(n\right)}\right)$$ is a subsequence, and we usually denote it just as $$ {a}_{\sigma \left(n\right)}$$