Trigonalizable/Flag via generalized eigenspaces/Exercise

Let

$\phi :V⟶V$

be a trigonalizable endomorphism, and let

$V=H_1\oplus \cdots \oplus H_m$

denote the direct sum decomposition into generalized eigenspaces in the sense of fact. Show that there exists a $\phi$-invariant flag $V_i$ such that, in this flag, the linear subspaces

$H_1,H_1\oplus H_2,\dots ,H_1\oplus \cdots \oplus H_j$

for $j=1,\dots ,m$ occur.

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