Nilpotent endomorphism/Dimension jumps/Exercise

Let

$\varphi \colon V\longrightarrow V$

be a nilpotent endomorphism on a finite-dimensional ${}K$-vector space ${}V$. Let

${}V_{i}:=\operatorname {kern} \varphi ^{i}\,.$

Show that for the dimension jumps, the relation

${}\dim _{K}{\left(V_{i}\right)}-\dim _{K}{\left(V_{i}-1\right)}\geq \dim _{K}{\left(V_{i+1}\right)}-\dim _{K}{\left(V_{i}\right)}\,$

holds.

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