How was Zeno's paradox solved using the limits of infinite series? This is a not necessarily the exact paradox Zeno is thought to have come up with, but it's similar enough:
A man (In this photo, a dot 1) is to walk a distance of one unit from where he's standing to the wall. He, however, is to walk half the distance first, then half the remaining distance, then half of that, then half of that and so on. This means the man will never get to the wall, as there's always a half remaining.
This defies common sense. We know a man(or woman, of course) can just walk up to a wall and get to it.
My calculus book says this was solved when the idea of a limit of an infinite series was developed. The idea says that if the distances the man is passing are getting closer and closer to the total area from where he started to the wall, this means that the total distance is the limit of that.
What I don't understand is this: mathematics tells us that the sum of the infinite little distances is finite, but, in real life, doesn't walking an infinite number of distances require an infinite amount of time, which means we didn't really solve anything, since that's what troubled Zeno? 
Thanks.

If the subject walks at a constant speed, you don't need calculus to calculate exactly when he will arrive at his destination. Just use the simple speed-equals-distance-over-time formula. This formula was unknown to the ancient Greek philosophers. They had no notion of actually measuring speed. It wouldn't be for another thousand years after Zeno's time that Galileo would formulate: 
$$s=\frac d{t}$$ 
Today, we know that in going from point $A$ to another point $B$, we will pass through infinitely many other points along the way. If we associate an event with our arrival at each of these points, then infinitely many events will have occurred in the interval. The modern mind has no trouble with this notion.
