There is an easy proof to show that for a finite object of any size $H^0(M) = |M|$, its 0-Hausdorff dimension is equal to its size. This is done by covering the set by sufficiently small balls and showing that one needs $|M|$ balls to cover it.
However, I'm asked to state this result in general. So $M$ may be countably infinite or uncountable.
How does one formalize the reasoning for finite objects to arbitrary sized objects? I was told to do it by first proving the case of countable sets and then extend it to unions of countable sets. I was told that this is enough to have it for arbitrary sets. Is this a general pattern in mathematics? Why is it true?