Upper triangular matrix/Basis/Invariant subspaces/Exercise

Let ${}\varphi \colon V\rightarrow V$ be a linear mapping on a ${}K$-vector space ${}V$. Let ${}v_{1},\ldots ,v_{n}$ be a basis of ${}V$, such that ${}\varphi$ is described, with respect to this basis, by an upper triangular matrix. Show that the linear subspaces

$\langle v_{1},\ldots ,v_{i}\rangle$

are ${}\varphi$-invariant

for every ${}i$.

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