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.