<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] Let $$A=\frac12\cdot\frac34\cdot\frac56\cdot...\frac{2n-1}{2n}$$
$$B=\frac23\cdot\frac45\cdot\frac67\cdot...\frac{2n}{2n+1}$$
Then $$A<B$$
Then $$A^2<AB=\frac1{2n+1}$$
$$A=\frac12\cdot\frac34\cdot\frac56\cdot...\frac{2n-1}{2n}<\frac{1}{\sqrt{2n+1}}$$
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