$\lim\limits_{n\to \infty} \frac{1}{n}\cdot \big((m+1)(m+2) \ldots(m+n)\big)^{\frac{1}{n}}$ where $m$ is a fixed positive integer is?

$$\lim_{n\to \infty} \frac{1}{n}\cdot \big((m+1)(m+2) \ldots(m+n)\big)^{\frac{1}{n}}$$

where $m$ is a fixed positive integer.

Here is my attempt:

According to Cauchy's theorem of limit if $\lim\limits_{n\to \infty}a_n=l$ then $\lim{(a_1a_2 \ldots a_n)}^{\frac{1}{n}}=l$

hence $\lim\limits_{n\to \infty}\frac {m+n}{n}$$\Rightarrow \lim\limits_{n\to\infty}\left(1+\frac{m}{n}\right)=1$

I'm 90 percent clear that my solution is correct. If not then please give me the right solution.