<s>[INST] How to get to this equality $\prod_{m=1}^{\infty} \frac{m+1}{m}\times\frac{m+x}{m+x+1}=x+1$? 
How to get to this equality $$\prod_{m=1}^{\infty} \frac{m+1}{m}\times\frac{m+x}{m+x+1}=x+1?$$

I was studying the Euler Gamma function as it gave at the beginning of its history, and need to solve the following product operator, tried a few things, but I could not ...
$$\prod_{m=1}^{\infty} \frac{m+1}{m}\times\frac{m+x}{m+x+1}=\prod_{m=1}^{\infty} \frac{m^2+mx+m+x}{m^2+mx+m} =\prod_{m=1}^{\infty} \frac{m(m+x+1)+x}{m(m+x+1)}$$
WolframAlpha checked in and the result really is correct.
http://www.wolframalpha.com/input/?i=simplify+prod+m%3D1+to+inf+%5B%28m%2B1%29%2Fm+*+%28m%2Bx%29%2F%28m%2B1%2Bx%29%5D
 [/INST] $$\begin{align}
\prod_{n=1}^N\frac{m+1}{m}\frac{m+x}{m+x+1}&=\left(\prod_{n=1}^N\frac{m+1}{m}\right)\left(\prod_{n=1}^N\frac{m+x}{m+x+1}\right)\\\\
&=\left(\frac{2}{1}\frac{3}{2}\frac{4}{3}\cdots \frac{N}{N-1}\frac{N+1}{N}\right)\left(\frac{1+x}{2+x}\frac{2+x}{3+x}\frac{3+x}{4+x}\cdots \frac{N-1+X}{N+x}\frac{N+x}{N+1+X}\right)\\\\
&=\frac{N+1}{1}\frac{1+x}{N+1+x}\\\\
&\to 1+x \,\,\text{as}\,\,N\to \infty
\end{align}$$
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