A poset has the descending chain condition if and only if every nonempty subset has a minimal element

Let P be a nonempty set partially ordered by \leq. Then the following are equivalent.

  1. Every nonempty subset of P contains a minimal element.
  2. If a : \mathbb{N} \rightarrow P is a sequence in P such that a_i \geq a_{i+1} for all i, then there exists n \in \mathbb{N} such that for all i \geq n, a_i = a_n. (The descending chain condition.

(1) \Rightarrow (2) Suppose every nonempty subset of P contains a minimal element, and let a be a descending chain in P. In particular, \{p_i\}_\mathbb{N} is a nonempty subset of P, which contains a minimal element a_n. We have a_n \leq a_i for all i as desired.

(2) \Rightarrow (1) Suppose P satisfies the descending chain condition, and let S \subseteq P be a nonempty subset. Suppose S does not have a minimal element. Define a descending sequence a : \mathbb{N} \rightarrow S inductively as follows. Let a_0 be any element of S. (We can do this since S is not empty.) Given that a_i is defined, since a_i \in S is not minimal, there exists a_{i+1} < a_i. Now a is a descending chain, and for all i, a_i > a_{i+1}; a violation of the descending chain condition.

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