The following bounds can be found in the German wikipedia page of Central Binomial Coefficient $$\tag{$\star$}\label{star} \frac{1}{2} \frac{4^n}{\sqrt{\pi n}}<\binom{2n}{n}< \frac{4^n}{\sqrt{\pi n}}, \quad n\ge 1. $$ However, no proof nor reference is provided. So my question:
Is there a simple way to derive the bounds in \eqref{star}?
References are also very welcome.