At the entrance exam of a french school, the following problem was given :
Characterize well-ordered subsets of $\mathbb{R}$
The only property I found was that such a subset must be at most countable, I do not know if this was what they expected or not, and I wondered if anyone had any other non-trivial characterization of such subsets ?
I noticed that a well-ordered subset $A$ of $\mathbb{R}$ must follow the following condition :
$$ (1)\quad\forall x\in \mathbb{R}, \exists \varepsilon > 0, ]x, x+\varepsilon[\cap A = \emptyset$$
Which can easily be found by noticing that you can't find a strictly decreasing sequence of elements of $A$.
From that property, we can deduce the countability of $A$ by taking $x\in A$ in (1) and picking a rationnal $r$ in $]x, x+\varepsilon_x[$.
It is also possible to associate to each $x\in A$ (except possibly $\max A$ if it exists) an element $S(x)\in A$ such that $S(x) > x$ and for all $y\in A$, $y >x\implies y\ge S(x)$.
As a last note, this exercice is meant for second year undergraduate students, but I am interested in all information about that problem !
Best Answer
Your condition (1), together with
indeed characterizes well-ordered subsets.
You already proved one direction.
For the other direction, assume (1) and (2), and let $H\subseteq A$ be arbitrary nonempty, and set $x:=\inf H$.
By condition (2), $x$ is finite ($x\ge\min A$), and condition (1) together with the definition of infimum imply $x\in H$.