Bounded sequence - Knowlpedia

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.

\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:

\inf_n x_n > -\infty \quad \text{and} \quad \sup_n x_n < \infty.

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