Proving finite second moment given $\sum_{n=1}^\infty nE(|X|1_{n<|x|\le n+1}) <\infty$

Question

I am given that $\sum_{n=1}^\infty nE(|X|1_{n<|X|\le n+1})<\infty$ and I am trying to prove or disprove that $E(|X|^2)<\infty$. I have been attempting to replicate a tail probability argument similar to what one does to find that $E(X)=\sum_{n=1}^\infty nP(X=n)$ to prove this, but I have not been getting very far. Can anyone provide a nudge in the right direction? Side note: This is not the exact problem I was given, just a twist on it that I am exploring in order to solve the real problem.

Answer 1

Hint:

1. Show that $\mathbb{E}(1_{n<|X| \leq n+1} |X|^2) \leq (n+1) \mathbb{E}(1_{n< |X| \leq n+1} |X|) \leq 2n \mathbb{E}(1_{n< |X| \leq n+1} |X|)$.
2. Conclude from $$\mathbb{E}(X^2) = \sum_{n \in \mathbb{N}} \mathbb{E}(1_{n< |X| \leq n+1} |X|^2)$$ and step 1 that $$\mathbb{E}(X^2) \leq 2\sum_{n \in \mathbb{N}} n \mathbb{E}(1_{n<|X| \leq n+1} |X|)<\infty.$$