I'm not sure you only need that many steps. As you start out, each step gives you a lot of distance. But log gets flatter and flatter the more steps you take.

If you knew the number of steps, I think you could only make a lower bound on spacing, not an upper bound. Since at worst all the steps could be clustered really close, except for the last one.

But now I have to think more because it seems like it should actually help the argument that the spacing gets finer and finer as you step, because that is what we want anyway. And is the number of steps actually important to the proof?