>>10fixed lower bound on distance
That is a separate and entirely legitimate question. If your distance function is either at least LOW or exactly 0, so that there are no distances in the open interval (0,LOW), then your convergent series can only be of a very particular form. For a[i] to converge to LIM, for any delta there must be a starting N over which dist(a[i>N],LIM)<delta. Pick delta=LOW/2 and you get a starting N over which the distances are under LOW and therefore 0, so a[i>N]=LIM. So the only convergent series are those that after a finite number of terms switch to a constant and never deviate again. So you get craziness like picking some nonzero X and having a[i]=X/2**i not being convergent. Your space only consists of isolated points and its interior is empty. So you won't be taking any nonconstant limits, like those needed for derivatives, and you don't get any proper calculus.