A random variable \(\xi\) has the density function

\[ p(x)= \begin{cases}0 & \text { for } x \leqslant 0 \\ \frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{\alpha-1} e-\beta_{x} & \text { for } x>0\end{cases} \]

Prove that as \(\alpha \rightarrow \infty\) the distribution of the variable \(\frac{\beta \xi-\alpha}{\sqrt{\alpha}}\) converges to the normal distribution with parameters \(a=0, \sigma=1\).