Let $W$ be a subspace of a vector space $V$ and $f$ be an endomorphism of $V$. Show that $$X = \{U \subseteq W\,|\,U \text{ is an $f$-invariant subspace of V}\}$$ with the partial order $\subseteq$ always has a maximal element.
My idea is to find the supremum of $X$, which is the maximal element I want to find, but I am not sure how to approch this problem generally.