If $Ax<x$, then $A^n \to 0$

Question

If $Ax<x$ where $A \in M_n({\mathbb R})$ nonnegative matriz and $x$ is a column matrix positive, show that:

$$A^n \to 0, \mbox{when } n \to \infty$$

I can obtain that the digaonal's coefficient are smaller that one, but not necessarily the rest. Help me, please.

Answer 1

Hint. Prove by mathematical induction that there is some $0<c<1$ that makes $A^nx\le c^nx$ for every $n\ge1$.