  We give a simple geometric proof that $e$ is irrational, using a construction
of a nested sequence of closed intervals with intersection $e$. The proof leads
to a new measure of irrationality for $e$: if $p$ and $q$ are integers with $q
> 1$, then $|e - p/q| > 1/(S(q)+1)!$, where $S(q)$ is the smallest positive
integer such that $S(q)!$ is a multiple of $q$. We relate this measure for $e$
to a known one and to the greatest prime factor of an integer. We make two
conjectures and recall a theorem of Cantor that can be proved by a similar
construction.
