Is there any proof of surjectiveness of matrix exponential using a density argument?

Question

i showed that $$\begin{array}{ccccc} f & : & \mathcal{M}_n(\mathbb{C}) & \to & \mathbb{GL}_n(\mathbb{C}) \\ & & A & \mapsto & \exp(A)=\sum\limits_{k=0}^{\infty} \frac{A^k}{k!} \\ \end{array}$$

is a surjective application with the Jordan blocks. But i was wondering if there was a proof using the density of diagonalizable matrices in $\mathcal{M}_n(\mathbb{C}) $.

Thank you.