Subsequence

Let $(x_n)_{n\in\mathbb{N}}$ be a sequence in a set $X$. A subsequence of $(x_n)$ is a sequence of the form $(x_{n_k})_{k\in\mathbb{N}}$, where $(n_k)_{k\in\mathbb{N}}$ is a strictly increasing sequence of natural numbers:

Subsequences capture partial asymptotic behavior and are indispensable in compactness arguments (e.g., Bolzano–Weierstrass) and in defining lim sup and lim inf .

Examples:

• From $x_n = (-1)^n$, the subsequence $x_{2k}=1$ is constant, and the subsequence $x_{2k+1}=-1$ is constant.
• If $x_n=1/n$, then $(x_{n_k})$ with $n_k=2k$ is the subsequence $1/(2k)$.
• Not every selection yields a subsequence: choosing $n_k=k$ gives the original sequence; choosing indices that do not increase strictly does not define a subsequence.