Spoiler:
Yes, he'll get there eventually. Because the track stretches equally along its length, the percentage Achilles has covered is not changed by each stretch. After a stretch he then runs another 10 metres, increasing his percentage covered, before the track stretches again.
After one second, he's run 10m, which is 10% of the track. The track then stretches to 200m, so the son of Thetis is now at the 20m mark (10 * (200/100)), still at 10%.
After two seconds, he's at 30m, 15% of the track. The track stretches to 300m, taking Achilles to 45m (30 * (300/200)), still 15%.
After three seconds, 18.33%.
After four seconds, 20.83%.
And so on, gradually approaching 100%.