Proof of a matrix norm inequality

Question

I have been struggling with how to prove the following:

$$\begin{align} \left\|A^k\right\| \le \left\|A\right\|^k \end{align}$$

I can prove it for $k=2$, and so I expect induction to work, but I have no idea how to carry it out.

Answer 1

In general, you expect this to hold for norms that are submultiplicative, i.e. $\|AB\|\leq\|A\|\,\|B\|$. Then $$ \|A^{k+1}\|=\|A^kA\|\leq\|A^k\|\,\|A\|\leq\|A\|^k\,\|A\|=\|A\|^{k+1}. $$