<s>[INST] Prove $ \frac 1 2 \cdot \frac 3 4 \cdot \frac 5 6 \cdots \frac{2n-1}{2n} < \frac 1 {(2n+1)^{0.5}} $ . Prove
$$
\frac 1 2 \cdot \frac 3 4 \cdot \frac 5 6 \cdots \frac{2n-1}{2n} < \frac 1 {(2n+1)^{0.5}}
$$
Can this be done by induction using the pi function.
If no, why not. 
 [/INST] Hint. If you want to prove it by induction, you may observe that, for $n\ge1$,
$$
(2n+2)^2=4n^2+8n+4>4n^2+5n+3=(2n+1)(2n+3)
$$ giving
$$
\frac1{2n+2}<\frac1{\sqrt{2n+1}\cdot \sqrt{2n+3}}, \qquad n\ge1,
$$ which is equivalent to the inductive step:
$$
\frac1{\sqrt{2n+1}}\cdot \frac{2n+1}{2n+2}<\frac1{\sqrt{2n+3}}.
$$
</s>