Bounded sets and sequences

Let $(X,d)$ be a metric space and let $A\subseteq X$ .

The set $A$ is bounded if there exist $a\in X$ and $r>0$ such that

A\subset B(a;r),

i.e., $A$ is contained in some ball.

A sequence $(x_n)$ in $X$ is bounded if its range $\{x_n:n\in\mathbb N\}$ is bounded.

Examples

In $\mathbb{R}$ , any interval $[a,b]$ is bounded; $\mathbb{R}$ itself is not bounded.
In a normed space with metric $d(x,y)=\|x-y\|$ , boundedness means $\|x_n\|\le M$ for some $M$ and all $n$ .
In the discrete metric, every subset is bounded (take radius $r=2$ around any point).