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 ?

Induction is a really efficient way for proving that
$$\frac{(2n-1)!!}{(2n)!!} = \frac{(2n)!}{4^n n!^2} = \frac{1}{4^n}\binom{2n}{n}<\frac{1}{\sqrt{2n}}\tag{1}$$
but it isn't the only option. For instance, we may notice that the LHS of $(1)$ can be written as $$\prod_{k=1}^{n}\left(1-\frac{1}{2k}\right)=\sqrt{\prod_{k=1}^{n}\left(1-\frac{1}{k}+\frac{1}{4k^2}\right)} \tag{2}$$
hence:
$$ \prod_{k=1}^{n}\left(1-\frac{1}{2k}\right)^2 = \frac{1}{4}\color{blue}{\prod_{k=2}^{n}\left(1-\frac{1}{k}\right)}\prod_{k=2}^{n}\left(1+\frac{1}{4k(k-1)}\right)=\frac{1}{4n}\prod_{k=2}^{n}\left(1+\frac{1}{4k(k-1)}\right)\tag{3} $$
since the blue term is a telescopic product. Since $1+z<e^{z}$ for any $z>0$, we also have:
$$ \prod_{k=1}^{n}\left(1-\frac{1}{2k}\right)^2 < \frac{1}{4n}\exp\color{purple}{\sum_{k=2}^{n}\frac{1}{4k(k-1)}}=\frac{1}{4n}\exp\left(\frac{n-1}{4n}\right)<\frac{e^{1/4}}{4n} \tag{4}$$
since the purple sum is a telescopic sum. By switching to square roots it follows that
$$ \frac{(2n-1)!!}{(2n)!!}=\frac{1}{4^n}\binom{2n}{n}< \color{red}{\frac{e^{1/8}}{2\sqrt{n}}}\tag{5}$$
that is a stronger inequality. If we directly compute $\prod_{k=2}^{+\infty}\left(1+\frac{1}{4k(k-1)}\right)=\frac{4}{\pi}$ through Wallis product, with the same approach we may also prove the sharper
$$\boxed{ \frac{(2n-1)!!}{(2n)!!}=\frac{1}{4^n}\binom{2n}{n}< \color{red}{\frac{1}{\sqrt{\pi\left(n+\frac{1}{4}\right)}}}.}\tag{6}$$
