Doubt in derivation of a proof in Information Theory
For fixed $p$, the central limit theorem shows that
$$
\begin{align*}
\lim_{n\to\infty} \Pr[\mathrm{Bin}(n,p) \leq np] &amp;= \lim_{n\to\infty} \Pr[\tfrac{1}{n}\mathrm{Bin}(n,p) \leq p] \\ &amp;=
\Pr[N(p,\tfrac{p(1-p)}{n}) \leq p] \\ &amp;= \frac{1}{2}.
\end{align*}
$$
In this particular case you can get quantitative bounds in several ways, for example using the Berry–Esseen theorem or large deviation theorems such as the Chernoff bound.