How to prove that a fly will turn around indefinitely?

Question

Mathematics

Two trains move towards each other, their speeds are 20 km/h, together with the first train a fly flies out at a speed of 40 km/h. The distance between the starting points is 100 km. As soon as the fly meets one of the trains, it turns around. How many times will the fly turn around before the trains meet?
I'm trying to find a rigorous mathematical proof, but so far there is only code that proves that it will unfold indefinitely.

Answer 1

Mathematically (when the fly is represented by a dot) it is infinite. You can prove it by writing down a series of distances between trains at the moments of a turn and showing that the members of this series do not turn to zero, although they tend to it.
Physically, as soon as the trains approach a distance equal to the size of a fly, it can no longer turn around.

Answer 2

From the contrary. Suppose a fly at some point reflected for the last time (and this is before the trains meet). It flies faster than a train and will reach another train faster than the second train and will be reflected there again. For the distance between trains, although it will become smaller, will still be greater than 0. But we assumed that this was the last reflection. Contradiction.
And we still have to prove why there can't be a last reflection when the trains meet. Let's say both trains and the fly met at the same point. And what happened at the time of the previous reflection? some non-zero distance between trains. But if you lose from now on, then the next reflection will be until the moment the two trains meet. Again a contradiction.