Proof
Using the Bernoulli variables \(Y_k\) each with mean p and variance p(1-p), note that the Central Limit Theorem applied to \(\overline{X} = \frac{\sum Y_k}{n}\) gives that
\begin{equation*}
\frac{\overline{X}-p}{\sqrt{p(1-p)/n}}
\end{equation*}
is approximately standard normal. By multiplying top and bottom by n yields
\begin{equation*}
\frac{\sum Y_k - np}{\sqrt{np(1-p)}}
\end{equation*}
is approximately standard normal. But \(\sum Y_k\) actually is the sum of the number of successes in n trials and is therefore a Binomial variable.