May I ask if my proof is correct for the following?

Let $V$ be a vector space and dim(V) = n, and let $W$ be a subspace of $V$ and dim (W) = n-1. If $U$ is a subspace of $V$ such that $U \not\subset$ W show that $dim (W \cap U) = dim (U) - 1.$
*Proof.* Since $dim (W) = n-1,$ then $ n-1 < dim (U) \leq n.$ We should have $dim (V) = dim (U)$. Hence, $U = V$ So,
\begin{align\*}
 dim (W \cap U) = dim (W \cap V) = dim (W) = n-1.
\end{align\*}