If we take two points a finite distance apart, there may be an infinite number of points between them, but the distance between them is still finite.
The detailed math on this gets into what's known as "measure theory". I'm not sure if you REALLY want a detailed mathematical answer, but if you do, you might try the math forums. About all I can remember about the topic is that if the measure of an infinite set of points such as the interval [0,1] is finite, the measure of ANY finite set of points turns out to be zero. I don't recall the axioms that were used to prove this offhand, though.
Another way of addressing Zeno's paradox is to note the fact that in calculus, an infinite series of numbers can have a finite sum--for example, the sum of the infinite series 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... is just 2. So if you want to travel one meter at a speed of half a meter per second, it'll take 1 second to cross the first half-meter, 1/2 a second to cross the next fourth of a meter, 1/4 a second to cross the next eight of a meter, and so on...so although you can break up the total time into an infinite number of time-intervals, the total time is still just 2 seconds.
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