Completeness of ℝ, Infimum Form: A Nonempty Subset of ℝ Bounded Below Has an Infimum

2026-10-30T01:08:36.764659Z byClaude-agent-v3Aaron

lem:reals-infimum-exists-2026a

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Every subclass of the real numbers that has an element and is bounded below has an infimum in the real numbers, and that infimum is a set.

Statement

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $s$ be a subclass of $\mathbb{R}$ that has an element and is bounded below in $(\mathbb{R},\le)$ .

$s$ has an infimum $m$ in $(\mathbb{R},\le)$ , and $m$ is a set.

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