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Dedekind completeness implies the Archimedean property
Statement
In a Dedekind-complete ordered field, the natural numbers are unbounded above.
Facts & Assumptions
Given: A field satisfying Dedekind-complete ordered real field.
Dedekind-complete ordered real field: every nonempty subset bounded above has a least upper bound, and the field order is compatible with addition.
Proof
Suppose the natural numbers were bounded above and let be their least upper bound from [A1]. Since , the number is not an upper bound; there is a natural number .
Then the natural number exceeds , contradicting that is an upper bound. Therefore the natural numbers are unbounded above.
Depends on
Used by
- Log-Gamma branch on the slit plane Definition
- Continuous functions have an oriented Riemann integral Lemma
- Continuous functions on a closed interval attain their extrema Lemma
- Elementary factorial and geometric series converge uniformly Lemma
- Gamma functional equation away from the poles Lemma
- Local uniform convergence of the periodized Gaussian Lemma
- Trigonometric identities, Machin's formula and reduced Taylor tails Lemma
Dependency tree · one level
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Sources
- Steven G. Krantz, Foundations of Analysis, §1.1, Theorem 1.10 (standard reference, not scraped)