2,500 years ago, the Greek philosopher Zeno made this argument: "Even if swift Achilles chases a tortoise, by the time Achilles reaches the tortoise's starting point, the tortoise has already moved a bit further ahead. Since this repeats forever, Achilles can never catch the tortoise." It sounds logical, but in reality, Achilles overtakes the tortoise with ease. So what goes wrong?
Zeno's mistake was assuming that "adding infinitely many terms always makes the result infinitely large." But if the values you're adding keep shrinking, the sum can add infinitely many terms and still stop at a finite value. You can split the time it takes Achilles to catch the tortoise into infinitely many tiny pieces, but the sum of all those pieces is itself finite.
Let's say the whole distance is 1. First cover half (1/2), then half of what's left (1/4), then half of what's left after that (1/8)... if you keep closing half the remaining gap forever, do you really never arrive?
Pick a ratio r, and see what value you approach when adding infinitely many numbers that keep shrinking by that ratio. As long as the ratio is less than 1, the sum always converges to a finite value.
What happens if you add 1 + 1/2 + 1/3 + 1/4 + ..., where the denominator increases by one each time? Even though each term shrinks, this sum grows without bound — just extremely, extremely slowly.
Yes, it's been completely resolved mathematically. But it took over 2,000 years. Zeno stated this paradox in the 5th century BC, and it wasn't formally solved until Newton and Leibniz invented calculus in the 17th century, followed by Cauchy and Weierstrass rigorously defining the concept of a "limit" in the 19th century.
The key to the resolution is this: Zeno assumed that "going through infinitely many steps must take infinitely long." But an infinite number of steps doesn't mean the time it takes is infinite. As we saw above, adding up distances that keep shrinking — 1/2, 1/4, 1/8... — forever, the sum stops at exactly 1. Time gets split up the same way, so you can complete infinitely many steps within a finite amount of time. "Infinitely many steps" and "infinitely long" turned out to be two different things.
Philosophically, though, some questions still linger. Whether space and time can really be divided endlessly, or whether there's some smallest possible unit (like the Planck length in physics), is still an active topic of research in physics today. Mathematics has proven that "the numbers work out even when you divide infinitely," but whether the universe actually operates that way is a separate question.