@rilla
That's a good point. Our brain tends to interpret it as a sequence rather than a condition. So it probably confuses us for the same reason that it confuses us that 1/3 is 0.3333... and 3/3 is 0.999... and 0.999... = 1 because it's not like it's forever trying to get to 1, it's actually just 1.
The proof that there are infinitely more real than natural numbers is very neatly explained in the video I posted, you just have to sit through it :P