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**undefined** N is a positive integer I suppose? If I'm understanding you correctly, then yes, your definitions work, except that you didn't mention that the set has a least element, for which A(n-1) is not defined.

Ordered sets are usually referred to as tuples. (Ordered pair is another name for 2-tuple, an ordered triple is a 3-tuple, etc.)

Note that since you're *given* the subset of the reals rather than trying to *construct* the set, you could use a simpler definition. These following definitions are equivalent:

Let $\displaystyle S \subseteq \mathbb{R}, |S|=N,N\in\mathbb{Z},N>0$. Let the N-tuple $\displaystyle X=(a_1,a_2,\dots,a_N)$ be an ordering of the elements of $\displaystyle \displaystyle S$.

Definition 1: $\displaystyle \displaystyle X$ is in sequential order $\displaystyle \iff \forall i,j\in\{1,\dots,N\}: i < j \iff a_i < a_j$.

Definition 2: $\displaystyle \displaystyle X$ is in sequential order $\displaystyle \iff \forall i,j\in\{1,\dots,N\}: i < j \implies a_i < a_j$.

Definition 3: $\displaystyle \displaystyle X$ is in sequential order $\displaystyle \iff \forall i,j\in\{1,\dots,N\}, i < j:a_i < a_j$.