Evaluation of Limit involved in the proof of Asymptotic Unbiasedness of S

Question

We know that $S^{2}$ is an unbiased estimator of $\sigma^{2}$ and $S$ is a biased estimator of $\sigma$. But if $n\rightarrow\infty$, then $S$ is an asymptotically unbiased estimator of $\sigma$. I found a proof here(first answer). But in the last step, instead of solving the limit, graphic visualization has been given. Can we somehow algebraically prove that bias$\rightarrow0$ as $n\rightarrow\infty$? I was trying to solve with Gamma function definition but couldn't find a suitable formula to apply here.

Answer 1

The following formula would be useful. $$ \Gamma(n+\frac{1}{2})=\frac{(2n)!\sqrt{\pi}}{2^{2n}n!}$$

Using this, you can show that $\sqrt{\frac{2}{n-1}} ~ \frac{\Gamma(n/2)}{\Gamma(\frac{n-1}{2})} \to 1$ as $n \to \infty$.

You can find a proof for the formula here.