If $||A^N|| < 1$ then is ||A||<1?

Question

following a previous question in functional analysis I asked

Let $X$ be a banach space and $A$ is a bounded linear operator on $X$, and there exists some natural $N$ for which $ ||A^N|| < 1 $. Is $ ||A||<1 $? Also does it hold that $ A^n \to 0 $?

We need to prove or give counterexamples in both parts

I cannot seem to prove either one but the thing is I cannot really think of some counterexamples and I dont know if any are true or not so I need help on both parts thanks all

Answer 1

Counterexample to the first question: A nilpotent matrix. The second part is true, however: Since $A$ is bounded linear there is a uniform bound $c$ on $||A^1||,\dots, ||A^n||$, and then we have $|A^{kn+i}v|\le c||A^n||^k|v|\to 0$.