Why the theory of limits does not solve Zeno's paradoxes

Zeno noted that in moving from point A to point B, first you have to move halfway to B. Then you have to move three quarters of the way to B. Then 7/8 of the way to B. And so on, in an infinite series. And he wondered how anyone can ever complete an infinite series of moves.

What the mathematical theory of limits shows is that, IF one completes that series, one will be at point B. Well, Zeno already knew that! The theory supplies a formal way of solving what an infinite series comes to in its limit. That is something completely different from answering Zeno's puzzle over how we can complete such an infinite number of moves!

UPDATE: To clarify, when we calculate the limit, what we do is figure out what the final result would be IF we were to do every addition in an infinite series. We don't actually do the additions, because that would take forever. But the job of a runner trying to cross the continuum between the starting line and the finish line is not to figure out where he would get to if he actually completed the infinite series of moves necessary to reach the finish line. His job is to actually reach the finish line, not to figure out how far away the finish line is!

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