Define the sequence S by

S(0) = 0
S(n+1) = 1 + S(n)

This is a sequence of natural numbers. This sequence does not converge, which means that the limit as n goes to infinite of S(n) is not a natural number (nor a real number for that matter).

You could try to write it as a function of time, S'(t) such that S'(1-0.5^n) = S(n). That is, S'(0)=0, S'(0.5)=1, S'(0.75)=2, etc. A possible formula is S'(t) = -log_2(1-t). You could then ask what is S'(1). The answer is that this is the same as the limit S(infinity), or as log(0), which are both not defined. So in fact S' is not a function from numbers between 0 and 1 inclusive to natural or real numbers, since the domain excludes 1.

You can similarly define a sequence of distributions over the natural numbers by

T(0) = {i -> 0.5 * 0.5^i}
T(n+1) = the same as T(n) except two values swapped

This is the example that you gave above. The sequence T(n) doesn't converge (I haven't checked, but the discussion above suggests that it doesn't), meaning that the limit "lim_{n->inf} T(n)" is not defined.