Deriving the Gaussian Distribution from the Sterling Approximation and the Central Limit Theorem

You can find anything on the web.  Earlier today, I noticed that the Sterling approximation and the Gaussian distribution both had a $\sqrt{2 \pi}$ in them, so I started thinking that maybe you could apply Sterling’s approximation to the binomial distribution with $p=1/2$ and large $n$ and the central limit theorem to derive the Gaussian distribution.  After making a few calculations it seemed to be working and I thought, “Hey, maybe someone else has done this.”  Sure enough, Jake Hoffman did it and posted it here.  The internet is pretty cool.