Problem: For $(X_n)_{n \geq 1}$ i.i.d. real RV with Var$(X_1)=1$ and $E(X_1)=0$ and $S_n$ denoting the partial sum of the RVs we have $$\lim_{n \to \infty} E(|S_n|)=\infty $$
My Approach: I have managed to show, thanks to the central limit theorem, that $\exists p >0$ such that for large enough $n \in \mathbb{N}$ (i.e. $n \geq n_0$) we have \begin{align}P (|S_n| \geq \sqrt{n}) \geq p>0, \ \forall n \geq n_0 \tag{*} \end{align}
I do want to use * to conclude the statement. My idea was now to use that for a positive RV $X$ we have $$E(X)= \int_0^\infty P(X \geq x) dx $$ However I am having trouble to connect this with my result (*) because evidently we have $P(|S_n| \geq \sqrt{n}) \geq P(|S_n| \geq n)$
I can write $$E(|S_n|) = \int_0^\infty P (|S_n| \geq x ) dx = \sum_{i=1}^\infty \int_{i-1}^i P(|S_n| \geq x)dx \\ \geq \sum_{i=1}^\infty \int_{i-1}^i P(|S_n| \geq i) dx = \sum_{i=1}^\infty P(|S_n| \geq i ) $$ I assume that I am on the wrong track. Maybe someone could provide me a hint to get me in the right direction again using (*) to conclude the statement in the problem.
Note that this might be what @Did suggest in the comments. However, since I haven't dealt with conditional probability/expectation yet I will rewrite it in my own words.
Note that for all $a>0$ we have $a \mathbb{1}_{|X| \geq a} \leq |X|$. Applying this to the problem at hand we easily see that \begin{align}\mathbb{E} (|S_n|) \geq \mathbb{E}( \sqrt{n} \cdot 1_{|S_n| \geq \sqrt{n}}) = \sqrt{n} \mathbb{E}(1_{|S_n| \geq\sqrt{n}}) = \sqrt{n} \mathbb{P}(|S_n| \geq \sqrt{n}) > \sqrt{n}p \end{align} which concludes the proof for $n \to \infty$.