Let $\{X_n \}$ be a sequence of integer valued i.i.d random variables that are symmetric around $0$, and $\mathbb{E}|X_1|^3<\infty, P[X_1 = 1]>0, P[X_1=0]>0$. Let $S_n = X_1+\dots+X_n$. Show that $$\lim_{n\rightarrow\infty}\sqrt{2\pi\sigma^2n}P[S_n=0]=1.$$
I know that I can write $$P[S_n = 0] =\frac{1}{2\pi} \int_{-\pi}^{\pi}\phi^n(t)dt=\frac{1}{2\pi\sqrt{n}}\int_{-\pi\sqrt{n}}^{\pi\sqrt{n}}\phi(t/\sqrt{n})^ndt.$$ Where $\phi$ is a characteristic function of $X_1$.
But apart from that I am stuck.
Hint:
$(\phi(t/\sigma\sqrt{n}))^n$ is the characteristic function if $\frac{S_n}{\sigma\sqrt{n}}$ which converges weakly to $N(0,1)$. Not worrying about the validity of getting the limit in side the integral, you get $$\begin{align}\sqrt{2\pi\sigma^2n}\frac{1}{2\pi}\int^{\pi}_{-\pi}(\phi(t))^n\,dt&=\sqrt{2\pi\sigma^2n}\frac{1}{2\pi\sigma\sqrt{n}}\int^{\pi\sigma\sqrt{n}}_{-\pi\sigma\sqrt{n}}\phi(t/\sqrt{n\sigma^2})^n\,dt\\&\quad\xrightarrow{n\rightarrow\infty} \frac{1}{\sqrt{2\pi}}\int^\infty_{-\infty}e^{-\frac{t^2}{2}}\,dt=1\end{align}$$
The existence of the third moment allows to find bounds for $(\phi(t/\sigma{n\sigma^2}))^n$ that allows you to use dominated convergence to justify passing to the limit inside the integral. See pages 136-140 of Durret, R., Probability Theory and Examples, 5th ed., Cambridge University Press, 2019. In particular p. 139