Mathematical induction problem: $\frac12\cdot \frac34\cdots\frac{2n-1}{2n}<\frac1{\sqrt{2n}}$ This is a problem that I tried to solve and didn't come up with any ideas
.?$$\frac{1}{2}\cdot \frac{3}{4}\cdots\frac{2n-1}{2n}<\frac{1}{\sqrt{2n}}.$$
All I get is $\frac{1}{\sqrt{2n}}\cdot\frac{2n+1}{2n+2}<\frac{1}{\sqrt{2n+2}}$ which evaluates to $1<0$Do you know what to do here ?

Prove by induction the stronger inequality: for any integer $n\geq 1$,
$$\frac{1}{2}\cdot \frac{3}{4}\cdots\frac{2n-1}{2n}<\frac{1}{\sqrt{2n+1}}.$$
The basic step is true: $1/2<1/\sqrt{3}$. 
Then in the inductive step you will obtain the inequality
$$\frac{1}{\sqrt{2n+1}}\cdot\frac{2n+1}{2n+2}<\frac{1}{\sqrt{2n+3}}$$
that is equivalent to
$$4n^2+8n+3=(2n+1)(2n+3)<(2n+2)^2=4n^2+8n+4$$
which holds.
