Convergence in Distribution using Moment Generating Functions

Question

Let $Y_{n} = \sqrt{\frac{1}{2}} \overline{X}$ where \begin{equation} \overline{X} = \frac{X_{1} + X_{2} + \cdots + X_{n}}{n} \end{equation} and the $X_{i}$ are independent. Use the moment generating functions technique to show that \begin{equation} Y_{n} \xrightarrow{d} N(0,1) \ \ \text{as } n \to \infty \end{equation} Where $\xrightarrow{d}$ means converges in distribution. I got this question in my first year exam and since then I've not been able to work it out. The distribution of the $X_i$ I guess shouldn't matter but I can't think of how to work this out otherwise.