if $U$ is compact, how show that $[U]_+$ is compact? Let $U$ compact in $\mathbb{R}^n$, then
$[a]_+=max\{0,a\},\;a\in\mathbb{R}$;
$[x]_+=([x_1]_+,...,[x_n]_+),\;x\in\mathbb{R}^n$;
$[U]_+=\{[x]_+:x\in U\}$
How show that $[U]_+$ is compact?
Thanks!

Hint: $[a]_+ = \frac{a+|a|}{2}$, which is a continuous function. Hence...
