Does this beginningless-past thought experiment result in several possible non-well-ordered sets?

Question

Assume there is an infinitely large universe with a beginningless past in which two eternal particles move at a constant speed of 1 km per “day” with respect to each other. Intuitively (to me), it seems that the distance between the two particles would, on any day, be infinite, since on any day there are an actual infinite number of prior days, and so the two particles would have already moved an actual infinite distance with respect to each other. However, it might be argued that the distance between the two particles could always be finite: On some day X the distance (in km) is 0, on day X-1 it is 1, on day X-2 it is 2, etc… So the ordered set of the distances between particles for every day up to day X would be (…, 4, 3, 2, 1, 0). Is this possible? If so, since it also seems possible that the this set could be (…, infinity, infinity, infinity), is the answer indeterminate?