Order is preserved by adding a constant and by adding inequalities

Depends on

• Ordered field

Used by

• Every nondegenerate interval of ℝ is uncountable 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 limit inferior is the least subsequential limit in ℝ̄ 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
• ⋂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 sequence with limsup = +∞: the greatest subsequential limit exists only in ℝ̄ 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
• 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 the domain {0} ∪ [1,2] every real is vacuously a limit at 0 Counterexample
• Over ℚ there is a nonconstant differentiable function with identically zero derivative, so Rolle and the mean value theorem both fail Counterexample
• ℝ covered by its closed singletons: every restriction of the indicator of {0} is continuous and the map is not, so the closed pasting lemma needs finiteness Counterexample
• ℝ is the union of a meager set and a set of measure zero, so smallness of category and smallness of measure are independent notions Counterexample
• The cover {(1/k, 1)} of (0,1) has no finite subcover, so (0,1) is not compact Counterexample
• The Dirichlet function on [0,1] has lower Darboux integral 0 and upper Darboux integral 1, so it is bounded and not Riemann integrable Counterexample
• The empty set is bounded and has no supremum Counterexample
• The identity from the cocountable topology on ℝ to the usual topology is sequentially continuous and not continuous Counterexample
• The indicator of ℚ has a limit at no point of ℝ Counterexample
• The indicator of the Smith-Volterra-Cantor set is discontinuous exactly on a nowhere dense set, and is not Riemann integrable, because that set does not have measure zero Counterexample
• Thomae's function is nonnegative, Riemann integrable on [0,1] with integral 0, and nonzero at every rational, so a vanishing integral does not force a nonnegative integrand to vanish Counterexample
• xₖ = (-1)ᵏ, yₖ = (-1)ᵏ⁺¹ give limsup(xₖ + yₖ) = 0 < 2 = limsup xₖ + limsup yₖ Counterexample
• xₖ = 1 + (-1)ᵏ, yₖ = 1 + (-1)ᵏ⁺¹ give limsup(xₖ yₖ) = 0 < 4 Counterexample
• xₖ₊₁ = xₖ + 1/xₖ from x₁ = 1 has strictly decreasing consecutive gaps and diverges, so no uniform c < 1 exists Counterexample
• ℤ and {n + 1/n : n ≥ 2} are disjoint closed subsets of ℝ at distance 0, so the set-to-set distance is not a metric Counterexample
• ℤ is closed and not compact, and (0,1) is bounded and not compact: neither hypothesis of Heine-Borel can be dropped Counterexample
• ψ(1/x) has no limit at 0: two sequences tending to 0 give values constantly 0 and constantly 1/2 Counterexample
• Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space Definition
• Limits at +∞ and -∞, and infinite limits at a point Definition
• The Cantor function on [0,1], defined on the Cantor set through ternary digits and extended constantly across each removed interval Definition
• The Cantor middle-thirds set as the intersection of the sets Cₙ obtained by removing open middle thirds Definition
• The extended real line ℝ̄ = ℝ ∪ {-∞, +∞}, its order, and the arithmetic that is left undefined Definition
• The Smith-Volterra-Cantor set: the same construction removing, at stage n ≥ 1, an open middle interval of length 4⁻ⁿ from each of the 2ⁿ⁻¹ remaining intervals Definition
• (-1)ᵏ has liminf = -1 and limsup = 1, so it does not converge Example
• (0,1) + (2,3) = (2,4), with supremum 4 = sup(0,1) + sup(2,3) Example
• (3x² - 1)/(x² + x) → 3 as x → +∞ Example
• ∫₀¹ x² = 1/3, computed from the Darboux definition with uniform partitions and the closed form ∑_k<n k² = n(n-1)(2n-1)/6 Example

…and 131 more results.

Dependency tree · two levels

2 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

• 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)