Vector space/C/Flag/R/Exercise

Let

${}0=V_{0}\subset V_{1}\subset \ldots \subset V_{n-1}\subset V_{n}=V\,$

be a flag in a ${}\mathbb {C}$-vector space ${}V$. We consider ${}V$ as a real vector space of real dimension ${}2n$. Show that there exist real linear subspaces

${}W_{i}\subseteq V\,$

such that

${}0\subset W_{1}\subset V_{1}\subset W_{2}\subset V_{2}\subset \ldots \subset V_{n-1}\subset W_{n}\subset V_{n}\,$

is a real flag.

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