More rigorous notation than "ellipsis" for "and so forth?"

Question

So I understand that in general, ellipsis are used in math where the pattern is easy enough to discern that it is up to the reader to understand what is meant by it. However, for such cases, I was wondering if anybody has any tricks to make such notations more rigorous? For example, what made this spring up in my mind was that I was trying to prove a generalized transitive law for inequalities. I'm working with axioms here, so if I'm bothering to be precise enough as to prove stuff from axioms then I might as well be precise about it, and I wanted a more rigorous way of stating

Let $\{a_1, a_2, \dotsm, a_n\}$ be a set of real numbers. Suppose $a_n > a_{n-1} > \dotsm > a_1$. Then $a_n > a_1$.

In typing this, I realized that I could use set builder notation to describe the set (i.e. "Let $\{a_k | 1 \leq k \leq n\}$..."$), but I still have no clue on what to do for the continued inequality short of creating my own, recursive notation.

Answer 1

If $i>j$, then $a_i> a_j$, or: let $a_i=f(i)$, where $f:[n]\to\Bbb R$ is an order preserving function. Here $[n]$ is a chain of $n$ elements represented by the numbers $1,2,\ldots,n$ and the usual ordering in $\Bbb N$. At any rate, I personally find nothing questionable in using three dots.

Answer 2

I think your notation is quite clear and rigorous enough. From what you wrote, it is quite clear to me that if $i < j$ then $a_i < a_j$ for any $1 \leq i < j \leq n$. I would suggest that if you have doubts about rigor, it would be better to add words rather than notation. For example:

Given a set $\{a_1, a_2, \ldots a_n\}$ consisting of $n$ distinct real numbers sorted in ascending order (so that the larger subscripts index larger elements), ...

Or maybe just

Given a set $\{a_1, a_2, \ldots a_n\}$ consisting of $n$ distinct real numbers sorted in ascending order, ...