Surjection from $M$ to the quotient $R/P$ for $R$ a local ring with maximal ideal $P.$

Question

Let $R$ be a Noetherian local ring with maximal ideal $P$ and $M$ a finitely generated module. I want to show that there is a surjection $M \rightarrow R/P$ with Nakayama's Lemma. We know that $M/PM$ is a finite dimensional vector space over the field $R/P.$ Hence, $M/PM \cong (R/P)^k$. Now there is a projection map $(R/P)^k \rightarrow R/P$. So composing each map with the natural surjection $M \rightarrow M/PM \rightarrow (R/P)^k \rightarrow R/P$, we obtain a surjection. However, this did not employ Nakayama's Lemma. Is this still correct?