How to denote the set $A$ correctly?

Question

Let be $I=\{1,2,\ldots, n\}$ is the set of indices. I have an increasing sequence of positive real numbers: $$0<a_1<a_2<\ldots <a_n.$$ Question. How to denote the set $A$ correctly?

My attemps are:

1. $A=\{a_i \in \mathbb{R}^+: a_i < a_{i+1}, i=1,2, \ldots,n-1\}.$
2. $A=\{a_i \in \mathbb{R}^+: a_i < a_{i+1}, i \in I\}.$
3. $A=\{a_i \in \mathbb{R}^+: a_i < a_{i+1}, i \in I\setminus\{n\}\}.$
4. $A=\{a_i \in \mathbb{R}_{>0}: a_i < a_{i+1}, i \in I\setminus\{n\}\}.$
5. $A=\{a_i \in \mathbb{R}_{>0}\}$ and $\forall i \in I\setminus\{n\}: a_i<a_{i+1}.$

Answer 1

If the $a_i $ are given (which is what I understand from your text), then you should simply write $A=\{a_i, i \in I\} $
The fact that the elements are ordered is another property which stands besides.
1 and 3 are correct in terms of indices, but they mean you are considering all such sets, and since any positive real belongs to such a set, it is in fact simply $\mathbb {R}^*_+$

Answer 2

Easy: Let $a_1,a_2,\ldots,a_n$ be an increasing sequence of positive real numbers. Many times explaining in words conveys equally well, and needs less expertise with LaTeX.