[HarryT]

Originally Posted by pdurrant

The right answer, but I'm not convinced that you've properly shown thar it actually is the right answer.

Spoiler:

For instance, if, instead of extending it 100m, would Achilles finish if they extended it by 100m, 200m, 300m, etc,? After all, he'll still be increasing the percentage of the track he's covered all the time.

The reason for it is...

Spoiler:

...that if Achilles is running at a constant speed, and the track is extending at a linear rate (any linear rate), his progress along the track will be some divisor of the arithmetic series:

1 + 1/2 + 1/3 + 1/4 + ...

and that series, as can trivially be shown, sums to infinity:

Group the terms as:

1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8), etc, doubling the number of terms each time. Each group sums to at least 1/2, so we have an infinite number of halves, which sums to infinity.

No matter HOW fast the track stretches linearly, therefore, Achilles will reach the end.