Sequence and natural numbers

Question

Why the domain of sequences has been chosen to be the set of natural numbers, and not for example the set of real numbers ?

Are there advantages from the fact that $\mathbb{N}$ is a countable set ?

Answer 1

The short answer is: because that's the way sequences are defined. But, in fact, the general concept of sequence is: a sequence is a function whose domain is an ordered set $(S,\preccurlyeq)$ which, as an ordered set, is isomorphic to $(\mathbb{N},\leqslant)$. So, the basic idea is that we can talk about the first element of the sequence, the second element of the sequence, and so on.

Answer 2

The name says it, in a sequence a term has a next term. In an uncountable universe, there are no next.