Prove by induction: inequality Use induction to prove the inequality:
$$\frac{1\cdot 3\cdot 5\cdots (2n-1)}{2\cdot 4\cdot 6\cdots 2n}\leq \frac{1}{\sqrt{n+1}}$$
I tried by multiplying $\frac{(2n+1)}{(2n+2)}$ to both sides but I don't know how to get $\frac{1}{\sqrt{n + 2}}$ on the RHS.

Hint:  For positive integers $n$ one has
$\frac{1}{\sqrt{n+1}}\cdot\frac{(2n+1)}{(2n+2)}\leq \frac{1}{\sqrt{n+2}}$
is true if and only if
$\frac{1}{n+1}\cdot \frac{4n^2+4n+1}{4n^2+8n+4}\leq \frac{1}{n+2}~~~~~$   (seen by squaring both sides)
is true if and only if
$(n+2)(4n^2+4n+1)\leq (n+1)(4n^2+8n+4)~~~~~$   (seen by cross multiplying)
