Show that $n \to \infty, \sqrt{n}(Y_{n}-p) \rightarrow N(0,p(1-p)$ 
Can anyone show me the correct working out to find the variance for $Y_n$, My variance seems to be $\frac{p(1-p)}{n}$

The variance is $\frac{p(1-p)}{n}$:
\begin{align*}
Var(Y_n) &= E\left((Y_n-E(Y_n))^2\right)\\
&=\frac{1}{n^2}E\left(\left[\sum_{i=1}^n\big(X_i-E(X_i)\big)\right]^2\right)\\
&= \frac{1}{n^2}\sum_{i=1}^n E\left((X_i-E(X_i))^2\right) \\
&= \frac{1}{n}\left[p(1-p)^2+p^2(1-p) \right]\\
&=\frac{p(1-p)}{n}.
\end{align*}
Then,
\begin{align*}
\frac{\sqrt{n}(Y_n-p)}{\sqrt{p(1-p)}}\rightarrow N(0, 1)
\end{align*}
in distribution, and, consequently,
\begin{align*}
\sqrt{n}(Y_n-p)\rightarrow N(0, p(1-p))
\end{align*}
in distribution.
