Interesting math problem

The rabbit family in modern society decided to compete with the tortoise family again in order to save the face of their ancestors who failed in the competition. In the afternoon, in order to show their strength, the rabbit family said that the tortoise should run 500 meters first (the whole journey is 2000 meters).

After receiving the gauntlet, the turtle family held a national meeting to discuss countermeasures. The tortoise can't win by strength, but it doesn't want to lose the face of its ancestors. Just then, a turtle appeared, claiming to outsmart his opponent.

Finally, the turtle replied: physical fitness is important in modern society, but intelligence is more important. We can beat you at the table. Why do you need a game? See you at the table if you have the courage!

The rabbit family thought that strength decided everything, and they were not afraid to serve, so they readily agreed!

At the table, the tortoise said, because of your arrogance, you lost the opportunity to save the face of your ancestors. The following is my discussion.

Let the speed of the tortoise be v 1 and the speed of the rabbit be v2. Because the rabbit runs 500m first, let the tortoise and the rabbit be so far apart, as shown in the figure.

a b c d

| - s0 - | - s 1 - | - s2 - | - s3 - |

Tortoise, tortoise, tortoise ...

Rabbit rabbit rabbit rabbit rabbit. ...

After the game starts, the tortoise plays A first, and then the rabbit runs. It takes time for the rabbit to run to A, and the tortoise to run to B within t 1. It takes time t2 for the rabbit to run to B and T2 for the rabbit to walk to C; It takes t3 for the rabbit to run to C, t3 for the tortoise to run to D, and so on. It can be concluded that the rabbit can only approach the tortoise infinitely, but can't catch up with the tortoise, so the rabbit can't win the tortoise.

The rabbit family thought for a long time, but found nothing wrong with the tortoise's reasoning. I regret letting the tortoise run 500 meters first. I regret that it is too late and I can only give up and go home.

But let's think about it. In reality, rabbits can catch up with turtles, but why? What's wrong with the turtle's reasoning?