<s>[INST] 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 ?
 [/INST] An easier proof without induction:
Let $r$ be a positive integer, whence we have by the AM-GM inequality,
$\dfrac{(2r-1)+(2r+1)}{2}>\sqrt {(2r-1)(2r+1)}$
$\Rightarrow \dfrac{2r}{2r-1}>\sqrt {\dfrac{2r+1}{2r-1}}$
$\Rightarrow \displaystyle\prod_{r=1}^n\left( \frac{2r}{2r-1}\right)>\prod_{r=1}^n\left(\sqrt {\frac{2r+1}{2r-1}}\right)$
$\Rightarrow \dfrac{2.4.6...(2n)}{1.3.5...(2n-1)}>\sqrt {2n+1}$
$\Rightarrow \dfrac{1.3.5...(2n-1)}{2.4.6...(2n)}<\dfrac{1}{\sqrt{2n+1}}<\dfrac{1}{\sqrt {2n}}. $
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