[Bill S]

Ah yes, that should have been "infinite."

Tricky stuff, this infinity. 

[JP]

Pete gave the example of 1/(distance remaining) as you move between two points becoming infinite.  Ligharrow gave the example of a series with infinite terms that sums to a finite number.

Absolutely right, undoubtedly mathematically sound, but has anyone adduced “objective physical evidence that such an event really happened”?

Now we're getting into the definition of "real," which is slippery.  But I can move from point A to point B, so the quantity 1/(remaining distance) diverges to infinity.  Similarly, the series is real--I can write it in notational form (though for obvious reasons I can't write it out explicitly). 

The problem is that there's a huge class of finite things which can't become infinite, so I agree with you for the most part.  But there's a smaller set of things that can, so I don't agree completely.