Power of operator norm

Question

Why is it the case that when A is a bounded linear operator on a Hilbert space that $||A^n||$ $\leq$ $||A||^n$? I found this question which is similar but not quite the same Finite Power of Operator Norm.

EDIT: Attached picture of the operator norm I am using.

Answer 1

Hint: $\|AB\| \le \|A\| \|B\|$

Answer 2

Hint : if $A,B $ are bounded linear operators, then

$$||ABx|| \le ||A||\cdot ||Bx|| \le ||A||\cdot ||B|| \cdot ||x||$$

for all $x $. Hence

$$||AB||\le ||A||\cdot ||B||. $$