In Stillwell's Mathematics and Its History we read
The need for clarification arose from the discovery of Harnack (1885) that any countable subset {x0, x1, x2, . . .} of R could be covered by a collection of intervals of arbitrarily small total length. Namely, cover x0 by an interval of length ε/2, x1 by an interval of length ε/4, x2 by an interval of length ε/8, . . . , so that the total length of intervals used is ≤ ε. (This is another proof, by the way, that R is not a countable set.)
I could not find a reference expounding such a proof, can anybody point to one?