$\mathscr{N}(S^{n_{0}})=\mathscr{N}(S^{n_{0}+1})$ for some $n_{0}$ $\implies\mathscr{N}(S^{n_{0}})=\mathscr{N}(S^{n_{0}+k})$ for all $k\in\mathbb{N}$.

Question

This is an exercise from Rudin.

Let $X$ be a Banach space, $T\in \mathcal{B}(X)$ be a compact operator, and $\lambda\ne 0$.

I want to prove that $\mathscr{N}((T-\lambda I)^{n_{0}})=\mathscr{N}((T-\lambda I)^{n_{0}+1})$ for some $n_{0}\in\mathbb{N}$ $\implies\mathscr{N}((T-\lambda I)^{n_{0}})=\mathscr{N}((T-\lambda I)^{n_{0}+k})$ for all $k\in\mathbb{N}$.

If this were for general $n\in\mathbb{N}$ then it would follow quite trivially via mathematical induction. However, I don't know how to proceed in this particular case.

It follows that the range of $T-\lambda I$ is the whole space and the dimension of the null space of $T-\lambda I$ is finite, but I don't think this is of much help.