Originally posted by Daemon Sin
At any given point during his infintely long journey our little worm friend has only ever travelled 1% of the distance along the rope.
This statement is not true, which is why I think you are confused on the problem. If it were true, then it's clear that the worm would never make it to the end.
But in this problem (one of my favorite problems because it is very counter-intuitive to me), we can show explicitly that the worm will make it to the end; furthermore, he will do so in a FINITE amount of time, ie., he doesn't need to live infinitely long -- only long enough.
The proof follows these lines:
Consider the fraction of rope that the worm has traversed after the Nth second. Since the stretching of the rope is uniform (ie, "affine" deformation), this fraction is preserved upon each stretch -- that is, the fraction is the same the instant before each stretch as it is the instant after each stretch.
After the first second the worm has traversed 1/100 of the rope. After the second second, the fraction is 1/100 + 1/200. After the 3rd second, the fraction is 1/100 + 1/200 + 1/300. And, generally, after the Nth second, the fraction is Summation(i=1 to i=N)[1/(100i)].
The question is then does this infinite sum ever get larger than 1? In fact, it must after finite N, because this sum is just 1/100 times the harmonic series, which is well-known to diverge (it blows up to infinity eventually). I think if you were to calculate at what N it actually exceeds 1, our worm would have to live longer than the universe has been in existence. Still, the important point is that he would only have to live a FINITE amount of time. Seems very counter-intuitive to me, but the math is what it is.
However, whether or not the worm gets to the end is not independent of how much we stretch the rope each time. If we stretch it sufficiently far each time, then we can make it so that the worm can never make it to the end, even if he lived infinitely long.
So a follow up question I would ask is: what is the minimum amount by which we need to stretch the rope each time such that the worm will never make it to the end of the rope in finite time?
The limiting case would be when the fraction of rope traversed by the worm converges to 1 as N goes to infinity.