I'm not really convinced with a proof, made by my teacher, of the following inequality (which is then used to proof the Law of Large Numbers with hypotheses of i.i.d random variables and existance of $E[|X|]$):
Let $X$ be a random variable, then: $\sum^{+\infty}_{n=1} \frac{1}{n^2} E[X^2.I_{\{|X|<n\}}] \leq 2E[|X|]$
The proof is quite simple, but I can't find the reason of a switch of two sums:
$$\begin{align} \sum^{+\infty}_{n=1} \frac{1}{n^2} E[X^2.I_{\{|X|<n\}}] &= \sum^{+\infty}_{n=1} \sum_{k \leq n} \frac{1}{n^2} E[X^2.I_{ \{k-1 \leq|X|<k\}}]\\ &\leq \sum^{+\infty}_{k=1}E[X^2.I_{ \{k-1 \leq|X|<k\}}] \sum^{+\infty}_{n=1} \frac{1}{n^2}\\ &\leq 2 \sum^{+\infty}_{k=1} \frac{1}{k}E[X^2.I_{ \{k-1 \leq|X|<k\}}]\\ &\leq \sum^{+\infty}_{k=1} E[|X|.I_{\{k-1\leq|X|<k\}}]\\ &=2E[|X|] \end{align} $$
I can't really understand why, passing from the second to the third term, he switched the sums: some hypoteses of absolute convergence should be needed, but don't find them. Thanks to everybody.