Trigonalizable/Flag via generalized eigenspaces/Exercise

Let

$\varphi \colon V\longrightarrow V$

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 ${}\varphi$-invariant flag ${}V_{i}$ such that, in this flag, the linear subspaces

$H_{1},H_{1}\oplus H_{2},\ldots ,H_{1}\oplus \cdots \oplus H_{j}$

for ${}j=1,\ldots ,m$ occur.

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