[mightytightywty]

Think of the ant's progress as a percentage. Every step he makes will increase his progress percentage across the rubber band. Therefore, if he keeps walking he will eventually make it to the other side. Am I missing something?

[lotharbot]

If the rubberband doubled in length every minute, the ant would still make progress in terms of percentage, but never come close to the end of the rubberband. He would approach the 2% point -- the sum of 1 + 1/2 + 1/4 + 1/8 + ...

In this case, he will eventually make it to the end -- the sum of 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... is divergent.

[mikeash]

That's necessary but not sufficient for the ant to make it to the end. Imagine an ant walking on a fixed rope but reducing its speed, such that during the first second it covers one quarter of the rope, the next second covers one eight, and reducing its speed by half like this each second. It will never reach the halfway point of the rope, let alone the end, but the percentage of the rope that it covers always increases.

In this particular case it does make it to the end, just showing how the fact that the percentage always increases is not quite enough to show it.

[mightytightywty]

But his speed would only slow down until he hits the 50% point. After that, his speed will increase until he hits the end.

[mikeash]

Huh? I am posing a different hypothetical problem to illustrate how covering a steadily increasing percentage of the total length doesn't necessarily imply that the ant ever reaches the end.