Suppose we have two sets of discrete events, $A$ and $B$. Then I think it is true that:

$$2\sum_{i \in A, j \in B}\Pr[i\ \textrm{AND}\ j] \leq \sum_{i \in A}\Pr[i]+ \sum_{j \in B}\Pr[j] +\sum_{i, j \in A}\Pr[i\ \textrm{AND}\ j] + \sum_{i, j \in B}\Pr[i\ \textrm{AND}\ j]$$

My intuition for why this should be true is just by analogy to the simple Cauchy-Schwarz like inequality for $a, b \in \mathbb{R}:$ $$2ab \leq a + b + a(a-1) + b(b-1)$$ To see why this identity is true, note: $$a + b + a(a-1) + b(b-1) = a^2 + b^2 = 2ab + (a-b)^2 \geq 2ab$$

Is my inequality true, and is there a simple proof?