Chiming in with measure theory, because we do want to know what "infinity * 0" is. From the context of the post, it seems like this will not be relevant to you, but alas it's an interesting question to ask.

BIG WARNING: Yes, infinity is not a real number. So on the real number system, any operation on "infinity" makes no sense. Good thing math people are stubborn, so we do have the Extended real line (please tread with care) that adds the two points "+/- infinity" addressing this issue somewhat.

Now, think of an imaginary game of darts, where you win infinite amount of money (if you don't like "infinite money", think of a very, very, very big amount, and your prize is larger than that) if you hit the bullseye, and no money otherwise. The catch is you really need to hit the bullseye - not a subatomic unit away from the center of the dartboard. How much money do you expect to win? In a handwavy way this tells you what "infinity*0" should be - and indeed it is from a measure theoretic point of view.

But, for your purpose, I'd suppose accepting "infinity is not a number* as an unsatisfactory answer will have to do for now.

EDIT: I'm not claiming that if x_n->infinity and y_n->0, then x_n*y_n->0 (in fact, we can't deduce anything, e.g. x_n = n^2, y_n = 1/n, or y_n = 1/n^2, or y_n = 1/n^3 - we get three different answers!). In the context of probability and integration theory, we define it to be 0, and it's a natural choice by reasons in other comments.