Thank you for writing a detailed explanation! I've slightly edited mine to explain the infinity jump. Yours is, of course, much more detailed.

For the infinitesimal number, I think it makes more sense to use {0}|{1,1/2,1/4,1/8,...} since it gets born at the same day as say 1/3. So it is easier to understand how it arises without "waiting" for all reals.