Subsequence
Definition
Suppose a sequence $\left\{ a_{n} \right\}$ is given. For a sequence of natural numbers $\left\{ n_{k} : n_{i} \lt n_{i+1} \right\}_{ k \in \mathbb{N}}$, we call $\left\{ a_{n_{k}} \right\}_{ k \in \mathbb{N}}$ a subsequence of $\left\{ a_{n} \right\}_{ n \in \mathbb{N}}$.
If a subsequence $\left\{ a_{n_{k}} \right\}$ converges, its limit is called a subsequential limit of $\left\{ a_{n} \right\}$.
Explanation
A subsequence is, quite literally, a sequence that is a part of the original sequence.
Related Theorems
Theorem 1: If $\lim\limits_{n \to \infty} a_{2n} = L$ and $\lim\limits_{n \to \infty} a_{2n+1} = L$, then $\lim\limits_{n \to \infty} a_{n} = L$.
Theorem 2:
Every bounded sequence in $\mathbb{R}^{n}$ has a convergent subsequence.