The Ant On An Elastic Rope

An ant starts to walk along an elastic rope which is 1 km long, at a speed of 1 cm per second (relative to the rope it is crawling on). While this happens the rope is uniformly stretched by 1 km per second (i.e. after 1 second it is 2 km long, after 2 seconds it is 3 km long, etc.). Assume the rope will never break and we have an infinite amount of time on our hands. Will the ant ever reach the end of the rope?

Here's a hint: Yes, yes it will.

If it were crawling along, say, an iron bar being lengthened, then no. The ant could obviously never reach the end. A speed of 1 cm/s is not enough to reach a target that is moving away at 1 km/s.

However, this is a an elastic rope. The intro said “relative to the rope it is crawling on”: that's your clue right there. As the rope stretches, the ant is carried along. To visualize this, imagine the ant not crawling but sitting still — at 10% the length of the rope, thus 100 meters from the beginning. Initially the rope is 1 km, a second later it's 2 km. Because of the rope's elastic properties, the ant is still at 10% except this is now 200 meters from the beginning.
Now visualize it has been crawling during this time. In that case, after the second the ant would be at 200 m plus 1 cm: 10.0005%. Hey, some small progress has been made! Keep this up long enough, and eventually 100% (the end of the rope) is reached. And yeah, that will get tougher the further it crawls. It's a very discouraging walk if you ask me.

In fact, people much smarter than me calculated how long it would take. It's not that bad, a mere 8.9 × 10⁴³⁴²¹ years ought to do it...
(Disclaimer: unfortunately for the ant it's unlikely the universe is still around by then. But points for trying.)