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$$ \left(1-\frac{1}{9}\right) \cdots \left(1-\frac{1}{n^2}\right) \left(1-\frac{1}{(n+1)^2}\right) = {n+2\over 2n+x}$$\\
The calculation holds for $n=2$ and for $n=k+4$.\\
Therefore it holds for $n \geq 2$. Q.E.D.

\textit{If you have that}
$$\prod_{n=2}^{\infty} \left( 1 - {1 \over n^2} \right) = {1\over 2}$$

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