Suprema and infima are unique

Depends on

• Complete ordered field (least-upper-bound property)
• Greatest lower bound (infimum)
• Ordered field

Used by

• Every nondegenerate interval of ℝ is uncountable Corollary
• Refuted: convergence uniformly on every compact subset of ℝ implies uniform convergence. The maps x ↦ x/(n+1) separate the two Counterexample
• Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space Definition
• Bounded variation and total variation on an interval Definition
• For bounded f on [a,b] and a partition P: the infimum mᵢ and supremum Mᵢ of f on the i-th subinterval, and the lower and upper Darboux sums L(f,P) = ∑ᵢ mᵢ Δᵢ and U(f,P) = ∑ᵢ Mᵢ Δᵢ Definition
• Jordan inner and outer content and Jordan measurable bounded sets in ℝᵐ Definition
• Lower and upper Darboux sums over a grid partition in ℝᵐ Definition
• The Cantor function on [0,1], defined on the Cantor set through ternary digits and extended constantly across each removed interval Definition
• The lower and upper Darboux integrals of a bounded f on [a,b] as sup_P L(f,P) and inf_P U(f,P), Darboux integrability as their equality, and the notation ∫ₐᵇ f Definition
• The lower and upper Darboux integrals over a nondegenerate rectangle in ℝᵐ Definition
• The order topology of a linearly ordered set, with the open rays as a subbasis; order-convex sets, order-density, the least upper bound property, and linear continua Definition
• C([0,1], ℝ) is complete, and on it the uniform metric and the supremum metric induce the same topology Example
• min(|x-y|, 1) on ℝ has the usual topology and diameter at most 1 Example
• The moving spikes on [0,1] converge pointwise to 0, do not converge uniformly, and do not converge in the topology of compact convergence Example
• The supremum metric d_∞(f,g) = supₓ |f(x) - g(x)| on the bounded real-valued functions on a set Example
• Convergence in the uniform metric is exactly uniform convergence: one N serving every point Lemma
• Epsilon characterisation of the supremum Lemma
• For a nonempty set X and a metric space (Y,d) the uniform metric ρ̄(f,g) = supₓ min{d(f(x),g(x)), 1} is a metric on Y^X Lemma
• min(d,1) and d/(1+d) are metrics uniformly equivalent to d, so every metric space carries a bounded metric with the same topology Lemma
• Monotonicity of the supremum under inclusion Lemma
• The supremum is attained exactly when a maximum exists Lemma
• The supremum metric d_∞(f,g) = supₓ |f(x) - g(x)| is a metric on the bounded real-valued functions on a nonempty set Lemma
• Total variation is additive over adjacent subintervals and decreases under restriction Lemma
• Conventions: sup ∅, unbounded sets, and the extended reals Remark
• A nested sequence of nonempty closed bounded intervals has nonempty intersection, and the intersection is a single point exactly when the lengths tend to 0 Theorem
• A nondecreasing sequence bounded above converges to the supremum of its range, and a nonincreasing sequence bounded below to the infimum Theorem
• A subset of ℝ is connected if and only if it is order-convex, that is, an interval Theorem
• Dini's theorem: on a compact metric space a nondecreasing sequence of continuous real functions converging pointwise to a continuous limit converges uniformly Theorem
• Every nonempty set bounded below has an infimum Theorem
• Every open subset of ℝ is a countable disjoint union of open intervals, namely its order components Theorem
• Existence and uniqueness of n-th roots: a unique a^1/n ≥ 0 with (a^1/n)ⁿ = a Theorem
• If (Y,d) is complete then Y^X is complete in the uniform metric, and so is C(X,Y) Theorem
• Integrable functions on [a,b] form a set closed under sums and scalar multiples, and ∫ₐᵇ(λ f+μ g) = λ∫ₐᵇ f + μ∫ₐᵇ g Theorem
• ℝ is uncountable (Cantor's nested intervals, 1874) Theorem
• The Cantor function is well defined, satisfies c(x) ≤ c(y) whenever x ≤ y, is surjective onto [0,1], and is constant on every interval removed from the Cantor set Theorem
• The long ray is a linear continuum, hence connected; every one of its at most countable subsets is bounded above, assuming countable choice Theorem

Cited to discharge well-definedness by Greatest lower bound (infimum).

Dependency tree · two levels

5 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

• Infimum and supremum (Wikipedia) (standard reference, not scraped)
• W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 1 (standard reference, not scraped)
• John K. Hunter, An Introduction to Real Analysis (standard reference, not scraped)