Let $X \sim B(n;\;p)$ and $\hat{p}=\frac{X}{n}$. I have to demonstrate that $Var(\hat{p}) \le \frac{1}{4}n$.
I started like, $$Var(\hat{p})=Var\left(\frac{X}{n}\right)=\frac{1}{n}Var(X)=\frac{1}{n}\left(np(1-p)\right)=p(1-p)$$
What can I continue the proof?
EDITED: as WimC noted, when you pull the $\frac{1}{n}$ out of the variance, it should become a $\frac{1}{n^2}$. Note that $p(1 - p)$ achieves its maximum at $p = \frac{1}{2}$; thus $p(1 - p) \leq \frac{1}{4}$. Putting it together, $$ \text{Var}(\hat{p}) = \frac{1}{n}p(1 - p) \leq \frac{1}{4n}. $$