I am working on this question which has three parts.
Let $X$ be a random variable with $X\geq 0$ and $E[X^{2}]<\infty$, and set $m_{i}:=E[X^{i}]$ for $i=1,2$.
(a) Prove that for every $0\leq a<m_{1}$, we have $P(X>a)\geq (m_{1}-a)^{2}/m_{2}.$
(b) Prove that $(E|X^{2}-m_{2}|)^{2}\leq 4m_{2}(m_{2}-m_{1}^{2})$.
(c) Show the following inequality, and compare it to (a) for $X=\sum_{k=1}^{n}\mathbb{1}_{A_{k}}.$ $$P\Big(\bigcup_{k=1}^{n}A_{k}\Big)\geq\sum_{k=1}^{n}P(A_{k})-\sum_{1\leq k<\ell\leq n}P(A_{k}\cap A_{\ell}).$$
I have proved (a) and the inequality in (c), but I got stuck in (b) and the second part of (c).
Below is my attempt for (b):
Set $Y:=X^{2}$, then $$Var(Y)=(E(X^{2}-m_{2}))^{2}=E(Y^{2})-(E(Y^{2}))^{2}=E(X^{4})-((E(X^{2}))^{2}=E(X^{4})-m_{2}^{2}.$$
However, it is still far from the desired result.
For (c), how could I apply the inequality to (a) as $X=\sum_{k=1}^{n}\mathbb{1}_{A_{k}}$?