If the random variable has a Bernoulli distribution $X_n\sim Bern(p)$, I want to find the distribution of $n\bar X$ ($\bar X$ is the average) and I want to prove that $\bar X_n$ converges to $p$ in $L^2$.
For the first part the distribution of the $nX_n$ is the binomial $Bin(n,p)$ and just out of curiosity I used the central limit theorem to prove $n\bar X\sim N(0,np^2(1-p)^2)$ when $n$ becomes larger.
For the second part, I've just expanded the square: $E(\bar X_n-p)^2=E(\bar X_n)^2-2pE(\bar X_n)+p^2\to 0$ because $E(\bar X_n)\xrightarrow{a.s.} 0$ by the strong law of the large numbers.
Am I right?
To prove that
$$\overline{X}_n \xrightarrow{L^2}p$$
you can use the iff condition (the proof of this condition is very easy):
$$\begin{cases} \lim\limits_{n \to \infty}\mathbb{E}[\overline{X}_n]=p \\ \lim\limits_{n \to \infty}\mathbb{V}[\overline{X}_n]=0 \end{cases}$$
In your case you have
$$\mathbb{E}[\overline{X}_n]=\frac{1}{n}np=p$$
$$\mathbb{V}[\overline{X}_n]=\frac{1}{n^2}npq=\frac{pq}{n}$$
...you are all set!