Proof that e is the following infinite fraction:

Question

How do I show

$e = 2+ 1/(1+1/(1+4+1/(1+...$?

Where for every 2ones in a row there is an even number (I am sure you all know what I am talking about).

Answer 1

I suggest the following paper by Henry Cohn:

A short proof of the simple continued fraction expansion of e

Abstract: This note presents an especially short and direct variant of Hermite's proof of the simple continued fraction expansion $e = [2,1,2,1,1,4,1,1,6,\dots]$ and explains some of the motivation behind it.

Take also a look through the references, especially "The simple continued fraction expansion of $e$" by C. D. Olds.