Proving the inequality $\frac12\frac34....\frac{2n-1}{2n}<\frac1{\sqrt{2n+1}}$ How to show this inequality:
$\dfrac{1}{2}\dfrac{3}{4}....\dfrac{2n-1}{2n}<\dfrac{1}{\sqrt{2n+1}}$
Using induction the inequality is verified for $n=1$
now assume that that the inequality holds for $n$,to show it for $n+1$
Then
$\dfrac{1}{2}.\dfrac{3}{4}....\dfrac{2n-1}{2n}\dfrac{2n+1}{2(n+1)}<\dfrac{1}{\sqrt{2n+1}}.\dfrac{2n+1}{2(n+1)}=\dfrac{\sqrt{2n+1}}{2(n+1)}<\dfrac{\sqrt{2(n+1)}}{2(n+1)}=\dfrac{1}{\sqrt{2(n+1)}}$
but I have to make it less than $\dfrac{1}{\sqrt{2n+3}}$ which is not coming.Any help

Another posibility is to use the fact that $$\frac{k-1}{k}<\frac{k}{k+1} $$ for any k greater or equal to 1. If you write this inequality for k=3,5,...,2n-1 and multiply them, the answer should occur. 
