<s>[INST] Prove $\frac{1}{2}\cdot\frac{3}{4}\cdot...\cdot\frac{2n-1}{2n}<\frac{1}{\sqrt{3n}}$ for all $n$. Prove for all $n$: $\frac{1}{2}\cdot\frac{3}{4}\cdot...\cdot\frac{2n-1}{2n}<\frac{1}{\sqrt{3n}}$.
Using induction, I tried the brain-dead method and went straight for $$\frac{2n+1}{2n+2}\cdot\frac{1}{\sqrt{3n}}<\frac{1}{\sqrt{3n+3}}$$ $$...$$ $$1<0.$$ After embarrassing myself, I looked around and I found this thread. Using induction, we can then easily prove $$\frac{1}{2}\cdot\frac{3}{4}\cdot...\cdot\frac{2n-1}{2n}\leq\frac{1}{\sqrt{3n+1}}$$ $$\frac{1}{2}\cdot\frac{3}{4}\cdot...\cdot\frac{2n-1}{2n}\leq\frac{1}{\sqrt{3n+1}}<\frac{1}{\sqrt{3n}}.$$ This gets me to the original problem. But in a problem solving standpoint, how do you think to use $\frac{1}{\sqrt{3n+1}}$? Is there some point in the first induction that leads to this idea? Or is there a better method than the above?
 [/INST] A First Approach
$$
\begin{align}
n\prod_{k=1}^n\left(\frac{2k-1}{2k}\right)^2
&=\frac14\prod_{k=2}^n\left(\frac{2k-1}{2k}\right)^2\frac{k}{k-1}\tag{1a}\\
&=\frac14\prod_{k=2}^n\frac{2k-1}{2k}\frac{2k-1}{2k-2}\tag{1b}\\
&=\frac14\prod_{k=2}^n\frac{\color{#C00}{k-1/2}}{\color{#090}{k}}\frac{\color{#75F}{k-1/2}}{\color{#C90}{k-1}}\tag{1c}\\
&=\frac14\color{#C00}{\frac{\Gamma(n+1/2)}{\Gamma(3/2)}}\color{#090}{\frac{\Gamma(2)}{\Gamma(n+1)}}\color{#75F}{\frac{\Gamma(n+1/2)}{\Gamma(3/2)}}\color{#C90}{\frac{\Gamma(1)}{\Gamma(n)}}\tag{1d}\\[3pt]
&=\frac1\pi\frac{\Gamma(n+1/2)^2}{\Gamma(n+1)\,\Gamma(n)}\tag{1e}\\[3pt]
&\le\frac1\pi\tag{1f}
\end{align}
$$
Explanation:
$\text{(1a)}$: pull the $k=1$ term out front and bring $n$ inside as a telescoping product
$\text{(1b)}$: rearrange terms
$\text{(1c)}$: divide numerator and denominator by $2$
$\text{(1d)}$: write the products as ratios of the Gamma function, using $\Gamma(x+1)=x\,\Gamma(x)$
$\text{(1e)}$: collect terms using $\Gamma(1)=\Gamma(2)=1$ and $\Gamma(3/2)=\sqrt\pi/2$
$\text{(1f)}$: $\Gamma(x)$ is log-convex
Thus, we get the stronger
$$
\prod_{k=1}^n\frac{2k-1}{2k}\le\frac1{\sqrt{\pi n}}\tag2
$$

A Slightly Simpler Approach with a Better Bound
$$
\begin{align}
\prod_{k=1}^n\frac{2k-1}{2k}
&=\prod_{k=1}^n\frac{(2k-1)2k}{4k^2}\tag{3a}\\
&=\frac1{4^n}\binom{2n}{n}\tag{3b}\\
&\le\frac1{\sqrt{\pi\!\left(n+\frac14\right)}}\tag{3c}
\end{align}
$$
Explanation:
$\text{(3a)}$: multiply numerator and denominator by $2k$
$\text{(3b)}$: $\prod\limits_{k=1}^n(2k-1)2k=(2n)!$ and $\prod\limits_{k=1}^n2k=2^nn!$
$\text{(3c)}$: inequality $(9)$ from this answer
In fact, using inequality $(9)$ from this answer, we get
$$
\frac1{\sqrt{\pi\!\left(n+\frac13\right)}}\le\prod_{k=1}^n\frac{2k-1}{2k}\le\frac1{\sqrt{\pi\!\left(n+\frac14\right)}}\tag4
$$
</s>