Well I was going to post a PDF showing the work I wrote up disproving it but I can't attatch PDF files, so here is my simplified explination. Oh, and it has been disproven by other people too, so it's not like this is ground breaking or anything
The faulty logic in Zeno's argument is the assumption that the sum of an infinite number of numbers is always infinite. While this seems intuitively logical, it is in fact wrong. For example, the infinite sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... is equal to 2. This type of series is known as a geometric series. A geometric series is a series that begins with one and then each successive term is found by multiplying the previous term by some fixed amount, say x. For the above series, x is equal to 1/2. Infinite geometric series' are known to converge (sum to a finite number) when the multiplicative factor x is less than one. Both the distance that Achilles travels and the time that elapses before he reaches the tortoise can be expressed as an infinite geometric series with x less than one. So, Achilles traverses an infinite number of "distance intervals" before catching the tortoise, but because the "distance intervals" are decreasing geometrically, the total distance that he traverses before catching the tortoise is not infinite. Similarly, it takes an infinite number of time intervals for Achilles to catch the tortoise, but the sum of these time intervals is a finite amount of time.