TITLE: Does closed imply bounded?
QUESTION [7 upvotes]: Definitions:1. A set $S$ in $\mathbb{R}^m$ is bounded if there exists a number $B$ such that $\mathbf{||x||}\leq B$ for all $\mathbf{x}\in S$, that is , if $S$ is contained in some ball in $\mathbb{R}^m$.2. A set in $\mathbb{R}^m$ is closed if, whenever $\{\mathbf{x}_n\}_{n=1}^{\infty}$ is convergent sequence completely contained in $S$, its limit is also contained in $S.$3. A set $S$ in $\mathbb{R}^m$ is compact if and only if it is both closed and bounded.

Does closedness not imply boundedness in general? If so, why does a compact set need to be both closed as well as bounded?

REPLY [18 votes]: The set $\{\,(x,y)\in\mathbb R^2\mid xy=1\,\}$ is closed but not bounded.
Even simpler, $\mathbb R^n$ itself is closed (but not bounded).

REPLY [2 votes]: $\mathbb{R}^m$ itself is a closed set. is it bounded?
But in case of Compact sets, they are closed as well as bounded in $\mathbb{R}^n$.