Let's say we decide to race on a track $1000$ km long. You are a $100$ times faster than me, meaning if we both start at the beginning, you obviously win. To make things more fair you give me a head start of $1$m. The distance is still very small, meaning you will obviously win.
A few premises:
-For you to win the race you need to overtake me
-To overtake me you need to reach a point of equivalence
-If there is no point of equivalence you can't beat me
Let's assume it takes you $1$ second to reach 1m. However in that $1$ second, I would have travelled a distance forward-lets say I am now at $1.01$m. You haven't caught up to me- I'm still $0.01$m ahead of you. It takes you $0.01$s to travel that $0.01$m. But in that $0.01$ s I would have travelled $0.0001$m, meaning I'm still ahead of you. Therefore you can never catch up to me- the distance between us will get infinitesimally small, but never $0$. Therefore since you can't catch up to me, you can never win.
This obvious paradox has been resolved through the fact that an infinitesimal series adds up to one- however, doesn't thid simply prove both people will finish the race? How does it prove the faster person will win?
Please don't simply give me a linear solution. I do not want to know when the faster person catches up – I want to know the mathematical flaw in the paradox's logic.
Best Answer
If you start 1 meter ahead of me, and it takes me 1s to reach your current position (apparently I run at 1 meter per second, and you run at .01 meters per second), $\frac{1}{100}$th of a second to reach your position at $t=1$, etc. I take $$1 + \frac{1}{100}+\frac{1}{100^2} + \cdots = \sum_{i=0}^{\infty}\frac{1}{100^i}$$ seconds to overtake you. Since $$\sum_{i=0}^{\infty}\frac{1}{100^i} = \frac{1}{1-\frac{1}{100}} = \frac{100}{99},$$ then after $\frac{100}{99}$ seconds, I will have overtaken you. This will occur well before we reach the finish line; we've only advanced $\frac{100}{99}$ meters (since apparently I run at 1 meter per second), and the finish line is more than $\frac{100}{99}$ meters from where we started. After I've overtaken you, I will be ahead at any further time.
The implicit error in the original claim that I cannot overtake you is the assumption that an infinite sum of positive quantities will necessarily be infinite. This has long been dealt with, and does not even require the use of infinitesimals.
Of course, it's possible for you to start so far ahead of me that I will only catch up to you when we get to the finish line; but that's hardly a paradox! Nor do I understand what your complaint is with "both people finish the race". Is there some problem with the slower person finishing after the faster one has?