Say I have a sequence of numbers - 1,2,3,4,5,6,7. I add 1 to 7 to create the next number in the sequence,8. The sequence is finite. I add 1 to 8 to get the next number in the sequence, 9. The sequence is finite. I keep on going... At what point does my sequence become infinite? How can my sequence ever become infinite?

I assume that there's some nonzero minimum time, however brief, that you require to perform each step of addition. In that case, you will never produce an infinite sequence of numbers: that is, there is no finite time at which you will have produced an infinite sequence of numbers. That fact doesn't imply that the positive integers aren't an infinite sequence of numbers -- only that you can't produce them in the described way in a finite amount of time.