Supremum vs. Maximum in the definition of the Lp norm

Question

The $L_p$ norm $||A||_p$ is defined as

$$\sup_{x \neq 0} \frac{||Ax||_p}{||x||_p} = \max_{||x||_p = 1} ||Ax||_p \tag{1}$$

I'm not quite getting why the LHS uses $\sup$ but the RHS uses $\max$. I know how both are defined but somehow I don't get it. The maximum requires the solution to be contained in the candidate set while the supremum does not. Thus, why does (1) hold?

Answer 1

In finite dimensional space (which is the case here) the unit sphere $\{x\in\Bbb R^n\mid ||x||_p=1\}$ is compact then the supremum of the continuous function $x\mapsto ||Ax||_p$ is attained.