Bounded sets and sequences

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

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

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

A sequence $(x_n)$ in $X$ is bounded if the set $\{x_n\mid n\in\mathbb{N}\}$ 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).