I mean, it's cleaner if you do it not for real numbers but rather for sets of whole numbers (which are of course in bijection with the reals). Then you can off

I mean, it's cleaner if you do it not for real numbers but rather for sets of whole numbers (which are of course in bijection with the reals). Then you can offload all such fiddliness to that one bijection, rather than dealing with it repeatedly in different contexts.