Exercise 4, § 1

a) Now prove that one can associate uniquely with every ordered set E an element of $\{ -\infty \} \cup \mathbf{N} \cup \{ +\infty \}$, denoted dev(E) (deviation of E), in such a way that the following properties are satisfied: α) We have $\operatorname{dev}(E) = -\infty$ if and only if the relation $a \leq b$ in E implies $a = b$. β) Let n be a positive integer. The deviation of E is $\leq n$ if and only if, for every infinite decreasing sequence $(a_k)_{k \in \mathbf{N}}$ of elements of E, one has $\operatorname{dev}([a_k, a_{k-1}]) \leq n - 1$ for all sufficiently large k. b) In order that $\operatorname{dev}(E) \leq 0$, it is necessary and sufficient that every strictly decreasing sequence of elements of E be stationary. We have $\operatorname{dev}(\mathbf{N}) = 0$, $\operatorname{dev}(\mathbf{Z}) = 1$, $\operatorname{dev}(\mathbf{Q}) = +\infty$. c) If E and F are ordered sets and $f : E \to F$ is a strictly increasing mapping, we have $\operatorname{dev}(E) \leq \operatorname{dev}(F)$. We have $\operatorname{dev}(E \times F) = \sup(\operatorname{dev}(E), \operatorname{dev}(F))$. d) Let S(E) be the set of infinite increasing stationary sequences of elements of E, ordered by the product order. Then we have $\operatorname{dev}(S(E)) = \operatorname{dev}(E) + 1$.