Bounded sequence

A sequence $(x_n)$ in a metric space $(X, d)$ is bounded if its range $\{x_n : n \in \mathbb{N}\}$ is a bounded set .

In $\mathbb{R}$

A real sequence $(x_n)$ is bounded if there exists $M > 0$ such that $|x_n| \leq M$ for all $n$.

Equivalently, $(x_n)$ is bounded if it is both bounded above and bounded below:

Key results

• Every convergent sequence is bounded (but not conversely).
• Every bounded sequence in $\mathbb{R}^n$ has a convergent subsequence (Bolzano-Weierstrass ).
• Every bounded monotone sequence in $\mathbb{R}$ converges.

Examples

• $((-1)^n)$ is bounded but not convergent.
• $(1/n)$ is bounded and convergent.
• $(n)$ is unbounded.