Finding a specific $T$-invariant subspace that satisfies 3 criteria

Question

Suppose that $V$ is an infinite-dimensional vector space over $F$, and $T : V \rightarrow V$ is linear. Suppose also that $W$ is a $T$-invariant subspace of $V$ . Show that there is a subspace $U$ of $V$ with the following three properties:

(a) $U$ is $T$-invariant;

(b) U\W = $0$; and

(c) for any $v\in V$ , there is a polynomial $p(x) \in F[x]$ such that $p(T)v\in U +W$.

My solution: define U=V\W then (b) (c) follow however I don't quite know how to prove that U is T-invariant (if it is).