The mistake is to treat infinity as a number that follows "normal" rules of arithmetic - it isn't and it doesn't.
To make matters even more confusing, consider that there are an infinite number of (irrational) numbers that lie between zero and one. This gets into the distinction between "countably infinite" (like the even and odd numbers, they can be put into a 1-to-1 correspondence with the "natural" numbers 1, 2, 3...) and "uncountably infinite" like the numbers between zero and one.
most of the great mathematicians who studied the properties of infinity (notably Georg Cantor) wound up spending time in mental institutions - and can you blame them?
More here:
http://en.wikipedia.org/wiki/Infinity