Proving an inequality or finding a counter example

I'm trying to prove that the following inequality holds for any $0\leq a_1,a_2,b_1,b_2\leq 1$:

$$|a_1a_2-b_1b_2|\leq |a_1-b_1|+|a_2-b_2|$$

Does anybody have an idea for proving this, or has a counter-example.

Thanks,

Hint: $$|a_1a_2-b_1b_2|=|a_1a_2-a_2b_1+a_2b_1-b_1b_2| \leq |a_1a_2-a_2b_1|+|a_2b_1-b_1b_2|$$