Why does this set a compact? Consider the following set in $\mathbb{R}^n$:

$$S = \{ x \ :\ \|x\| = 1 \}$$

Why is this set a compact? I don't know about closed, but it doesn't look bounded to me.
HINT: $S$ is the boundary of a certain ball.
$S$ is representing a boundary and it is an inverse image continuous function $||x||$
Compact Set is a subset of a topological space if for every open cover of there exists a finite subcover of.