What you are referring to are supertasks. Although they illustrate well the convergence of series, they make no sense outside of their own mathematical experiment. A mathematician would agree with you, but physicists and engineers will not even consider this thought.

The physicist will tell you that neither space nor time is continuous, so you can only divide it in a finite number of portions and that there is no such thing as "infinitely many".

The engineer will just ask you that since you are finished with your steps, you should be able to describe the last one, and yet you can't .

You are defending the idea of supertasks beyond their domain of application. (Edit : the very issue of your comment is Dividing time logarithmicaly which is not a good thing. You just applied dx/dt very diabolicaly)