How to prove that $\lim_{n \to\infty} \frac{(2n-1)!!}{(2n)!!}=0$ So guys, how can I evaluate and prove that $$\lim_{n \to\infty} \frac{(2n-1)!!}{(2n)!!}=0.$$ Any ideas are welcomed.
$n!!$ is the double factorial, as explained in this  wolfram post.

$\lim\limits_{n\to \infty}\dfrac{(2n-1)!!}{(2n)!!}=0\quad$ and $\quad\lim\limits_{n\to \infty}\dfrac{(2n+1)!!}{(2n)!!}=\infty.~$ Where it really gets interesting, however, 
is when we attempt to evaluate their product. Their polar-opposite tendencies will cancel each 
other out, yielding $~\lim\limits_{n\to \infty}\dfrac{(2n-1)!!}{(2n)!!}\cdot\dfrac{(2n+1)!!}{(2n)!!}=\dfrac2\pi~,~$ which is the famous Wallis product. 
See also Basel problem for more information.
