Consider $X_{1} \dots X_{n} \dots $ i.i.d variables with $\mathbb{E}X = 0$ and finite non-zero second momentum of $X$. We want to find limit of $\frac{\mathbb{E}|S_{n}|}{\sqrt{n}}$, where $S_{n}= \sum_{k=1}^{n} X_{k}.$
My approach : Central limit theorem give us : $\frac{S_{n}}{\sqrt{n}}$ converges in distribution to $N(0,1)$-standard distribution. Now there is a theorem that give us that there is exists a probability space such as $\frac{S_{n}}{\sqrt{n}}$ converges a.e.. So we have that in some probability space our variable is converges almost everywhere to standard distribution, so it converges to the same in probability (because convergence almost everywhere implies convergence in probability). Now we may use Vitali's theorem , which give us that convergence in $L$ is the same as convergence in probability, if $|\frac{S_{n}}{\sqrt{n}}|$ is uniformly integrable, which is obviously true, so the limit should be a zero (the mean value of standard distribution). Am I right?