Let $X_{1},X_{2},...$ be a sequence of independent and identically distributed random variables with $E[X_{i}]=\mu$ and $D^{2}[X_{i}]=\sigma^2$. Furthermore, let $F_{X}$ denote a cumulative distribution function of a random variable $X$ and $\Phi$ – the cumulative distribution function of the standard normal random variable. Finally, let $S_{n}=\sum_{i=1}^{n} X_{i}$ and $\mathcal{N}(\mu,\sigma^2)$ – normal random variable with parameters $\mu$, $\sigma^2.$
I want to prove that for every $\epsilon>0$ there exists $N$ such that for every $n>N$ and every $x$ $$\mid F_{S_{n}}(x)-F_{\mathcal{N}(n\mu,n\sigma^2)}(x)\mid <\epsilon.$$
My proof:
Fix $\epsilon>0$.
By the central limit theorem there exists $N$ such that for every $n>N$ and every $x$ $$\mid F_{\frac{S_{n}-n\mu}{\sigma \sqrt{n}}}(x)-\Phi(x)\mid <\epsilon.$$ So $$ \mid F_{S_{n}}(\sigma\sqrt{n}x+n\mu)-\Phi(x)\mid <\epsilon.$$ Now if we let $x'=\sigma\sqrt{n}x+n\mu$ $$ \mid F_{S_{n}}(x')-\Phi(\frac{x'-n\mu}{\sigma\sqrt{n}})\mid <\epsilon.$$ But $\Phi(\frac{x-n\mu}{\sigma\sqrt{n}})=F_{\mathcal{N}(n\mu,n\sigma^2)}(x)$, which concludes the proof.
Is this correct?