Evaluate $\lim_{n\to\infty}(1+\frac{1}{a_1})\cdots(1+\frac{1}{a_n})$, where $a_1=1$, $a_n=n(1+a_{n-1})$ 
Evaluate $\lim_{n\to\infty}(1+\frac{1}{a_1})(1+\frac{1}{a_2})\cdots(1+\frac{1}{a_n})$,
  where $a_1 = 1$, $a_n = n(1+a_{n-1})$

\begin{align*}
&\lim_{n\to\infty} \left(1+\frac{1}{a_1}\right)\left(1+\frac{1}{a_2}\right)\cdots\left(1+\frac{1}{a_n}\right) \\
&= \lim_{n\to\infty} \frac{1}{a_1} \cdot \frac{1}{2} \cdot \frac{1}{3} \cdots \frac{1}{n} \cdot (a_n + 1) \\
&= \lim_{n\to\infty} \frac{a_n + 1}{n!}.
\end{align*}
Then I'm stuck. How to proceed?

Show by induction that
$a_1=1!/0!$
$a_2=2!/0!+2!/1!$
$a_3=3!/0!+3!/1!+3!/2!$
...
$\color{blue}{a_n=n!/0!+n!/1!+n!/2!+...+n!/(n-1)!}$
Thereby the limit is $1/0!+1/1!+2/2!+...1/n!+...=e$.
