The sequence $\{\ker(f-\lambda Id)^{k} - \ker(f-\lambda Id)^{k-1} \}_{k}$ is decreasing

Question

Given $V$ a vector space over a field $\mathbb{K}$ and a string $[d_{1} < d_{2} < \cdots < d_{k} = n]$, where $d_{i} = \dim \ker(f-\lambda Id)^{i}$.

I know that $$\{0\} \subset \ker(f-\lambda Id) \subset \cdots \subset \ker(f-\lambda Id)^{k} = V.$$

I would like to prove that the sequence $$\{dim \ker(f-\lambda Id)^{k} - dim \ker(f-\lambda Id)^{k-1} \}_{k}$$ Is decreasing.

Any help or solution would be appreciated.