Exercise 4.7.1.2. Let $(S, \leq )$ be a partially ordered set. Show that the following conditions are equivalent:
- $(1)$
The partial order $\leq $ is well-founded: that is, every nonempty subset of $S$ contains a minimal element.
- $(2)$
The set $S$ does not contain an infinite descending sequence $s_0 > s_1 > s_2 > \cdots $.