Proof that $e$ is irrational. (Proof verification) Here, $e = \sum_{k=0}^\infty 1/k!$. Define $S_n$ as:
\begin{align}
S_n = n!e - n!\sum_{k=0}^n \frac{1}{k!}
\end{align}
where $n!\sum_{k=0}^n \frac{1}{k!}$ is an integer. Write $e = 1/0!+1/1!+...+1/n!+1/(n+1)!+...$
\begin{align}
S_n &= n!\left(e - \sum_{k=0}^n \frac{1}{k!}\right) = n!\left( \frac{1}{0!}+
\frac{1}{1!}+...+\frac{1}{n!}+\frac{1}{(n+1)!} +... - \sum_{k=0}^n \frac{1}{k!}\right)\\
&=n!\left( \frac{1}{0!}+
\frac{1}{1!}+...+\frac{1}{n!}+\frac{1}{(n+1)!} +... - \frac{1}{0!} -  \frac{1}{1!} - ... -  \frac{1}{n!}
\right)\\
&=n!\left( \frac{1}{(n+1)!}+\frac{1}{(n+2)!}+... \right) \\
&= \frac{1}{(n+1)}+\frac{1}{(n+2)(n+1)}+... < \frac{1}{(n+1)}+\frac{1}{(n+1)^2}+... = \frac{1}{n}
\end{align}
So, $0<S_n<1/n$. Now, if $e=p/q$ then we would have an integer between $0$ and $1$ for a large $n$.
$$0 < n!p - qn!\sum_{k=0}^n \frac{1}{k!}<\frac{q}{n}$$
Is this correct? 
 A: It's basically fine. To tie up the details, you should specify exactly how large $n$ needs to be.
Also, writing "$0<S_n =$ (something that isn't equal to $S_n$) is bad notation. I'm not even sure exactly what you were trying to say there. If you're not sure how to write what you mean in symbols, use some words to clear things up.
A: Your proof seems sufficient enough, as far as I can tell. You did make a slight typo in the final inequality (multiplying through by $q$ would mean you have $qS_n$ there); this was touched on in another answer and you already edited your answer to rectify this.
If I had to nitpick anything of significance...
Personally it could use a lot of elaboration to explain various manipulations and results - where they come from and so on. I found myself having to fill in the holes myself; it can make it a bit difficult for someone to see where you're going.
But this may just be a personal issue. If nothing else, at least everything seems to logically follow. But taking some time to explain everything for the reader will help. (In particular if this is a proof for a class or something: no sense in getting points taken off because you thought it was obvious what was going on.)
