Proof that an n-dimensional subspace over $\mathbf{C}$ is closed

Question

I'm trying to understand this proof in Rudin's Functional Analysis text:

I don't understand the line about $p$ being contained in the closure of $Y \cap tV \subset f(tB) \subset f(t \overline{B})$. We have that $p$ is in $\overline{Y}$ and $p$ is in $tV$, but why does this imply that $p \in \overline{Y \cap (tV)}$?