Order on the rationals

Depends on

• The rationals as equivalence classes of pairs of integers
• Arithmetic on the rationals
• Order on the integers

Used by

• Finite and infinite regular continued fractions Definition
• Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction Definition
• Order on the reals Definition
• Rational powers aʳ of a positive base Definition
• √· on [0,∞) is uniformly continuous and exactly 1/2-Hölder, and is not Lipschitz Example
• On [0,1] the function x^β is β-Hölder and is α-Hölder for no rational α > β, so the Hölder classes are strictly nested Example
• The integral test applied to ∑ 1/ι(k+1)ᵖ for rational p>0, cross-checked against the published p-series theorem Example
• FALSE: a^m/n := (a^1/n)ᵐ extends to negative bases False statement
• Every rational has a positive-denominator representative Lemma
• Laws of rational exponents Lemma
• Monotonicity of r ↦ aʳ and of a ↦ aʳ Lemma
• Rational powers do not depend on the representative Lemma
• The integers embed in the rationals Lemma
• The rationals are Archimedean Lemma
• Contraction implies Lipschitz implies uniformly continuous implies continuous; every Hölder map is uniformly continuous, and a Lipschitz map on a bounded space is Hölder for every exponent Theorem
• Hölder's inequality for finite sums (rational exponents) Theorem
• If |f(x) - f(y)| ≤ C|x-y|^α on an interval for some rational α > 1 then f is constant Theorem
• Minkowski's inequality for finite sums (rational exponent) Theorem
• The rationals form a totally ordered field Theorem
• Weighted AM-GM inequality with rational weights Theorem
• Young's inequality for products (rational conjugate exponents) Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• T. Tao, Analysis I, 3rd ed., §4.2 (standard reference, not scraped)
• L. S. Krapp, Constructions of the real numbers: a set theoretical approach (Oxford, 2014) (standard reference, not scraped)