Prove that as (n) tends to infinity [ frac{1}{Gammaleft(frac{n}{2} ight)} sqrt{left(frac{n}{2} ight)^{n}} int_{0}^{1+t} int^{frac{overline{2}}{n}} z^{frac{n}{2}-1} e^{-frac{n z}{2}}

Prove that as \(n\) tends to infinity

\[ \frac{1}{\Gamma\left(\frac{n}{2}\right)} \sqrt{\left(\frac{n}{2}\right)^{n}} \int_{0}^{1+t} \int^{\frac{\overline{2}}{n}} z^{\frac{n}{2}-1} e^{-\frac{n z}{2}} d z \rightarrow \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{t} e^{-\frac{1}{2} z^{2}} d z \]

Apply Lyapunov's theorem to the chi-square distribution.