Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

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 $.