>>11106865You mean as in the indexes? Why indexes for sequences/series are integers?
A sequence is really just a function , where X is just the space where the elements of the sequence reside. You could very well do something like or even . Why don't we do that? Well, think about why we define sequences in the first place. The naturals in the domain are really just indexes for each element, they don't mark a particular "length step", since the elements are already free to be whatever you want them to be or however far apart you want them to be.
Both Z and Q are countable sets. That means that there exists a bijection between N and Z, and also between N and Q, so using a composition between those bijections and the functions g,h defined above, defining a sequence using Q and Z is exactly the same as using N. The distance between the points (assuming the space has a metric to begin with) wouldn't even change, because all you are doing is changing their indexes, basically just renaming them. Why N, then? Because it's simply the "easiest" countable set to work with, we can use tools like mathematical induction, and in general the structure of the naturals is pretty well known.
Even if we used your idea, using steps of epsilon length as indexes, you could see that the first step would be one epsilon apart, the second two, and so on. In other words, would just be a bijection, and if we have a sequence with epsilon "length steps" , then using the composition we'd arrive again at a sequence that has N as its domain, which is just much easier to work with.
