Applying Central Limit Theorem to show that $E\left(\frac{|S_n|}{\sqrt{n}}\right) \to \sqrt{\frac{2}{\pi}}\sigma$ In the book Probability Essentials, by Jacod and Protter, the following question has bugged me for a long while and I'm wondering if it is bugged. The question is an application of Central Limit Theorem:
Let $(X_j)_{j\geq1}$ be iid with $E[X_j] =0$ and $\sigma_{X_j}^2 = \sigma^2 < \infty$. Let $S_n = \sum_{j=1}^n X_j$. Show that
$$ \lim_{n\rightarrow \infty} E\left\{\frac{|S_n|}{\sqrt{n}}\right\} = \sqrt{\frac{2}{\pi}}\sigma$$
What I tried and am aware of:
Interchanging E and lim is wrong as $\frac{|S_n|}{\sqrt{n}}$ does NOT converge a.s to $|Z|$ where $ Z \sim \mathcal{N}(0,\sigma^2)$, but does so in distribution. Note the use of continuous mapping theorem.
Weak convergence aka convergence in distribution implies $E[f(X_n)] \rightarrow E[f(X)]$ when $X_n \rightarrow^d X$ for f continuous and bounded. $f(x)=|x|$ is not bounded.
I tried using a truncated version of $|X|$. But at one stage I had to swap limits and couldn't justify the steps.
I appreciate any help. Also kindly avoid Skorokhod's theorem if possible as it has not been covered. Although if you have an idea using that, I'm all ears.
Note: The RHS is $E|Z|$.

We assume $\sigma=1$.


*

*Show that for all $K$ and all $n$, 
$$P\left(\frac{|S_n|}{\sqrt n}\geqslant K\right)\leqslant\frac 1{K^2}.$$ 

*This implies 
$$\int \frac{|S_n|}{\sqrt n}dP\leqslant \int_{|S_n|<K\sqrt n} \frac{|S_n|}{\sqrt n}dP+\frac 1{K^2}.$$

*Use the definition of weak convergence together with $x\mapsto \min\{|x|,K\}$, a continuous bounded map.



Actually, there is a deeper result (not needed here) called the invariance principle (see Billingsley's book Convergence of probability measures) which stated the following. Take $\{X_n\}$ a sequence of i.i.d. centered random variables, with $EX_n^2=1$, and define for each $n$, a random function $f_n(\omega,\cdot)$ in the following way:


*

*$f_n(\omega,kn^{-1})=\frac 1{\sqrt n}\sum_{j=1}^kX_j(\omega)$ for $0\leqslant k\leqslant n$. 

*$f_n(\omega,\cdot)$ is piecewise linear. 


The measures associated with these functions converge in law to a Brownian motion. 
Now, to get the result, take the continuous bounded functional $F\colon C[0,1]\to \Bbb R$ given by $F(f)=|f(1)|$. The first version of the invariance principle, found by Kac and Erdős, was about the functional $F(f):=\sup_{0\leqslant x\leqslant 1}f(x)$. Then Donsker generalized it. 
