Ordered field

Depends on

• Field

Used by

• A continuous real function on a compact subset of ℝ is bounded Corollary
• Every continuous function on an interval has a primitive; two primitives differ by a constant; and ∫ₐᵇ f = G(b)-G(a) for any primitive G Corollary
• Every nondegenerate interval of ℝ is uncountable Corollary
• For every ε > 0 in a complete ordered field there is a natural n ≥ 1 with 1/n < ε Corollary
• If f,g are integrable on [a,b] then so are | f|, f², fg, max(f,g) and min(f,g), and |∫ₐᵇ f| ≤ ∫ₐᵇ| f| Corollary
• If X is nonempty, some row fibre is at least the average size and some row fibre is at most the average size Corollary
• ℝ((t⁻¹)) does not have the least-upper-bound property; its canonical naturals have no supremum Corollary
• ℝ((t⁻¹)) has the nested interval property for lengths tending to 0 Corollary
• Stolz-Cesaro, 0/0 form: if bₖ is strictly decreasing to 0, aₖ → 0, and the difference quotient converges, then aₖ/bₖ converges to the same value Corollary
• The Cesaro matrix satisfies the Silverman-Toeplitz conditions, giving a second proof of the Cesaro mean theorem Corollary
• The reverse triangle inequality Corollary
• Under dependent choice, a continuous real-valued map on a closed subspace of a normal space extends to the whole space, and a map into an open interval extends into that same open interval Corollary
• ι(Dₙ) = ι(n) ι(Dₙ₋₁) + (-1)ⁿ for n ≥ 1, and Dₙ = (n-1)(Dₙ₋₁ + Dₙ₋₂) for n ≥ 2 Corollary
• [0,1) is neither open nor closed in ℝ Counterexample
• {0} ∪ [1,2] is closed, has an isolated point, and is not perfect Counterexample
• {q ∈ ℚ : q ≥ 0, q² < 2} is closed and bounded in ℚ and is not compact Counterexample
• ⋂ₖ (-1/k, 1/k) = {0} is not open Counterexample
• 1/4 lies in the Cantor set and is the endpoint of no removed interval, so the endpoints do not exhaust it Counterexample
• A field homomorphism of ordered fields need not preserve order Counterexample
• A function differentiable on [0,1] whose derivative is unbounded, hence not Riemann integrable Counterexample
• A function that is not Riemann integrable although | f| is Counterexample
• A list of six distinct reals with no strictly increasing sublist of length four and no strictly decreasing sublist of length three Counterexample
• A relation whose row fibres all differ from the average size, so the averaging principle gives a bound that no fibre meets exactly Counterexample
• A sequence with limsup = +∞: the greatest subsequential limit exists only in ℝ̄ Counterexample
• A summability matrix failing exactly one Silverman-Toeplitz condition and transforming a convergent sequence to a divergent one Counterexample
• A supremum need not belong to its set: sup(0,1) = 1 ∉ (0,1) Counterexample
• A three-set count that drops the triple intersection and returns the wrong answer Counterexample
• aₖ = (-1)ᵏ, bₖ = k have aₖ/bₖ → 0 while the difference quotient oscillates, so Stolz-Cesaro has no converse Counterexample
• An unbounded set has no supremum: the naturals inside ℝ Counterexample
• Continuous f and integrable sign-changing g with ∫ₐᵇ fg ≠ f(ξ)∫ₐᵇ g for every ξ Counterexample
• Continuous fₙ → 0 pointwise on [0,1] with ∫₀¹ fₙ = 1 for every n Counterexample
• For the Dirichlet function every uniform partition with rational tags gives Riemann sum 1, so the sums converge along that sequence of tagged partitions although the function is not integrable: the mesh condition of the Riemann definition quantifies over all tagged partitions and cannot be weakened to one sequence Counterexample
• In {0} ∪ [1,2] with the metric of ℝ, the closure of B(0,1) = {0} is {0} while the closed ball is {0,1} Counterexample
• In ℝ(t) the rationals are not dense: no rational lies strictly between 0 and 1/t Counterexample
• Integrable φ and integrable f with φ∘ f not integrable: the order of the hypotheses in the composition theorem cannot be reversed Counterexample
• Not every ordered field is Archimedean Counterexample
• Null times divergent has no rule: xₖ = 1/k with yₖ = ck gives product limit c, and with yₖ = k² gives divergence Counterexample
• On (0,∞) the metrics |x-y| and |1/x - 1/y| have the same topology and are not uniformly equivalent Counterexample
• On a closed interval of ℚ there is a continuous unbounded function, a bounded one with no maximum, and one without the intermediate value property Counterexample
• On ℝ the metrics |x-y| and min(|x-y|,1) are uniformly but not Lipschitz equivalent Counterexample

…and 317 more results.

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Sources

• W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 1 (standard reference, not scraped)
• M. Spivak, Calculus, 4th ed., Ch. 1 (standard reference, not scraped)
• University of Illinois Chicago notes: Ordered field axioms (standard reference, not scraped)