A problem on bounded invertible linear operator in Banach space

Question

Let $X$ be a Banach space. Let $T : X \to X$ be a invertible linear operator and $M > 0$ be such that $\|T^{-k}\| \le M$ for all $k \ge 1$. Prove that $\inf_ {n\ge1} \|T^n(x)\| > 0$ for all $x \ne 0$ in X.

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I am totally helpless. Can I get some help?

Answer 1

Since $\|T^{-k}\| \leq M$ for all $k \geq 1$, we have by definition $\|T^{-k}(x)\| \leq M\|x\|$. $\|x\| = \|id(x)\| = \|T^{-n}(T^{n}x)\| \leq \|T^{-n}\|\|T^n x\| \leq M\|T^n x\|$. It seems we are done.