Projection is continuous

Question

Let $V$ be a Banach space and let $W,Z$ be closed subspaces such that $V= W \oplus Z$. I have to show that the natural projection $P:V \to W$ is continuous. I want to use the generalization of the closed graph theorem. So let $(w_n + z_n)_n \in V$ be a sequence such that $w_n+z_n \to w+z \in V$. We may assume that $(w_n)_n$ converges to some $w' \in W$. We're done if we can show that $w=w'$. But that's the part where I'm stuck.