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?

From where you stuck you can proceed as
\begin{align}
\frac{a_n+1}{n!}&=\frac{a_n}{n!}+\frac{1}{n!}=\frac{1}{(n-1)!}\frac{a_n}{n}+\frac{1}{n!}=\frac{1}{(n-1)!}(a_{n-1}+1)+\frac{1}{n!}=\\
&=\frac{a_{n-1}}{(n-1)!}+\frac{1}{(n-1)!}+\frac{1}{n!}=\ldots=
1+\frac{1}{1!}+\ldots+\frac{1}{n!}.
\end{align}
