Probability inequality proof I'm stuck on a homework question and don't even know where to start. Here it goes: 
If A and B are two events which are not impossible, prove that $$P(A\land B)\times P(A\lor B)\le P(A)\times P(B)$$

In general if $d+a=c+b$ and $a\leq b\leq c\leq d$ then:
$$ad=\frac{1}{4}\left(d+a\right)^{2}-\frac{1}{4}\left(d-a\right)^{2}\leq\frac{1}{4}\left(c+b\right)^{2}-\frac{1}{4}\left(c-b\right)^{2}=bc$$ 
This as a direct consequence of:$$d-a\geq c-b\geq 0$$
This can be applied by taking $a=P(A\cap B)$, $b=P(A)$, $c=P(B)$ and $d=P(A\cup B)$
