As I mention in another comment, I am specifically referring to the fact that when we computer scientists or discrete mathematicians use the term
in proofs, we were more often interested in the "unbounded"ness than something with "actual" infinite cardinality... the fact that I understand the difference is precisely
why I tended to start saying unbounded more often. Using infinity to mean unbounded is really too powerful, and a proof should use the least power necessary. (It was only an intuitive feeling at the time; I have a much better understanding of that fact now.)
When infinity is the relevant concept, use it.