  We give an alternate proof of the existence of the asymptotic expansion of
the Bergman kernel associated to the $k$-th tensor powers of a positive line
bundle $L$ in a $\frac{1}{\sqrt{k}}$-neighborhood of the diagonal using
elementary methods. We use the observation that after rescaling the K\"ahler
potential $k\varphi$ in a $\frac{1}{\sqrt{k}}$-neighborhood of a given point,
the potential becomes an asymptotic perturbation of the Bargmann-Fock metric.
We then prove that the Bergman kernel is also an asymptotic perturbation of the
Bargmann-Fock Bergman kernel.
