TITLE: Let $B$ have distribution $\text{Binomial}(n,p)$. Then what is $ \lim_{n \to \infty} \mathsf P\left[ \frac{B}{n} < p\right]$
QUESTION [5 upvotes]: Let $B$ have distribution $\text{Binomial}(n,p)$. What is the tool for analysing $$ \lim_{n \to \infty} \mathsf P\left[ \frac{B}{n} < p\right]$$?  
I think the answer should be  $ \lim_{n \to \infty} \mathsf P\left[ \frac{B}{n} < p\right]=\frac{1}{2}$. My reasoning is that $np$ is roughly the median of $B$ hence the limit should be $1/2$. 
However, I would like to see a proper proof.

REPLY [4 votes]: We have by the De Moivre–Laplace theorem $$\mathsf P\left(\frac B n < p\right)=\mathsf P\left(\frac{B - n p}{\sqrt n\sqrt{p (1-p)}}<0\right)\xrightarrow{n\to\infty}\mathsf P(\mathcal N(0,1)<0) = \frac12.$$