> But this will always be a positive number, therefore in some finite number of seconds, the ant will cover 100% of the distance. Correct result, but that reas

> But this will always be a positive number, therefore in some finite number of seconds, the ant will cover 100% of the distance. Correct result, but that reasoning is dodgy. It's easy to imagine a different problem where an ant always covers a positive proportion of the rope every unit time, but the ant doesn't reach the end of the rope. E.g. construct a situation where, in the 1st unit of time, the ant covers a quarter of the rope, in the 2nd an eighth, in the 3rd a sixteenth, ..., in the nth a 1/2^{n+1}th. In that scenario, as time tends to infinity the ant asymptotically reaches the half-way point. The reason it works in the problem in the article is that, there, the additional proportions the ant covers in each unit of time form a divergent series (~the harmonic series), meaning the sum tends to infinity as n-> infinity. In my version above, they form a convergent series, meaning the sum tends to a finite number (in my example, 0.5) as n-> infinity. (Of course, a convergent series with a sum >1 would also mean the ant reaches the end of the rope in finite time).