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?
Best Answer
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).