Prove $\| A^n\|^{1/n}$ converges to a finite positive limit, where $A$ is a square matrix.

Question

I'm stuck at the following problem:

Let $A$ be a non-nilpotent, real square matrix of dimension $n$ and define a matrix norm by $\| A \| = \sup_{\|x \| = 1} \|Ax\| $, where $ x\mapsto \|x \| $ is a norm on $\mathbb R^n$. Prove that $ \| A^n\|^{1/n}$ converges to a finite positive limit.

What I have succeeded so far: I prove that $\lim_{n}\|A^n\|^{1/n} = \inf_{n \geq 1} \| A^n \|^{1/n}$, but I could not show $\|A^n\|^{1/n}$ is bounded away from 0 to make the infimum positive. Any help would be highly appreciated!