How do I prove $\lim_{n\to\infty} \frac{n^z n!}{z(z+1)\cdots (z+n)}=\frac{1}{z}\prod_{n=1}^\infty \frac{(1+\frac{1}{n})^z}{1+\frac{z}{n}}$? How do I prove $$\lim_{n\to\infty} \frac{n^z n!}{z(z+1)\cdots (z+n)}=\frac{1}{z}\prod_{n=1}^\infty \frac{(1+\frac{1}{n})^z}{1+\frac{z}{n}}$$?
This could be proven if $$\lim_{n\to\infty} \frac{\prod_{k=1}^n (1+\frac{1}{k})}{n} = 1$$ but I can neither prove this one.
Please help!

Hint:
\begin{align}
\lim_{n\to\infty} \frac{n^z n!}{z(z+1)\cdots (z+n)} &= \frac{1}{z} \lim_{n \to \infty} \frac{ \left( \frac{2}{1} \cdot \frac{3}{2} \cdot \frac{4}{3} \cdots \frac{n}{n-1} \right)^z n!}{(1+\frac{z}{1})(1 + \frac{z}{2}) (1 + \frac{z}{3}) \cdots (1 + \frac{z}{n}) \cdot n! } \\
&= \frac{1}{z} \lim_{n \to \infty} \frac{ \left\{ \left( 1 + \frac{1}{1} \right) \cdot \left( 1 + \frac{1}{2} \right) \cdot \left( 1 + \frac{1}{3} \right) \cdots \left( 1 + \frac{1}{n} \right) \right\}^z}{(1+\frac{z}{1})(1 + \frac{z}{2}) (1 + \frac{z}{3}) \cdots (1 + \frac{z}{n}) } 
\end{align}
using that as $n \to \infty$, $\frac{n}{n-1}, \frac{n+1}{n} \to 1$.
