The paradox of rabbit vs tortoise race

Hello Puzzle, I would like to test you with this famous paradox. Please try to solve it yourself and don’t look it up on the web.

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A rabbit and a tortoise decides to race. To keep things fair, the rabbit gives the tortoise a head start of say, 500 meters. When the race begins, the rabbit starts running at a speed much faster than the tortoise, so that by the time he has reached the 500-meter mark, the tortoise has only walked 50 meters further than him. But by the time rabbit has reached the 550-meter mark, the tortoise has walked another 5 meters. And by the time he has reached the 555-meter mark, the tortoise has walked another 0.5 meters, and so on. This deduction process can continue again and again over an infinite number of times while still remain true, there is no end to this chain process.

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Apparently, this seems to prove that the rabbit can never overtake the tortoise, whenever he reaches somewhere the tortoise has been, he will always still have some distance left to catch up. But in reality, we know the rabbit will overtake the tortoise. So now I ask you this : how to crack this paradox and defeat the false logic? I’m waiting for your answer.