Bounded sequence

Let $(X,d)$ be a metric space and let $(x_n)$ be a sequence in $X$. The sequence is bounded if its range $\{x_n:n\in\mathbb{N}\}$ is a bounded subset of $X$, i.e. if there exist $x_0\in X$ and $M\in[0,\infty)$ such that

Boundedness is a minimal compactness-type hypothesis: in $\mathbb{R}^k$, bounded sequences have convergent subsequences (Bolzano–Weierstrass ).

Examples:

• In $\mathbb{R}$, $x_n=\sin n$ is bounded (since $|\sin n|\le 1$).
• In $\mathbb{R}$, $x_n=n$ is not bounded.
• In a normed space, any convergent sequence is bounded.