Proving that every smooth manifold is metrizable

Question

I'm trying to prove that every smooth manifold is metrizable. The outline of my proof looks like this:

1. By the Whitney Embedding Theorem, we can embed any smooth manifold $M$ into $\mathbb{R}^{2n+1}$ for some $n$.
2. This embedded submanifold inherits a topology from $\mathbb{R}^{2n+1}$ generated by balls defined using the standard Euclidean distance metric. Thus, $M$ is metrizable.

Is that all that's needed? It almost seems too simple.