Here's an example where the inequality is strict. You can test your argument against this example, to pinpoint where you went wrong.
Let , . Thus a_n is alternately 1 and 3, b_n is alternately 3 and 1. The lim inf for both sequences is 1. But when you multiply the sequences together you get a sequence where each term is 3. So the lim inf of the product is 3.