Supremum
Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.
A limit that forgets to be continuous
Every one of the functions x, x², x³, … is as smooth as anything could be, and every column of the picture settles down. What they settle on has a jump in it — and the quantity that sees the difference is the largest gap anywhere, which is a number about the whole graph rather than about any point of it.
Reached from below, or not at all
Every limit ordinal anybody meets is the end of an increasing sequence — ω, ω·2, ω^ω, all of them approached one step at a time. The first uncountable ordinal is not, and the reason it is not constrains the size of the continuum.
The subsequence that has to exist
Every bounded list of numbers has a part that settles down. A bounded list of functions need not: the waves sin 2πkx never come within 1.76 of one another. One extra condition — that no member may change faster than a bound they all share — restores the guarantee, and it is the reason a differential equation with a continuous rule has a solution at all.
Which curves have a length at all
A length is defined as a supremum over inscribed polygons, which behaves because every refinement is longer than the last. It is also sometimes infinite — and the condition separating the two cases is a sum of absolute differences that either settles or does not.
![Polynomials climbing to the square root, each above the last. The iterates of p ↦ p + (x − p²)/2 at steps 1, 2, 3, 4, 6, 9 on [0, 1] beneath the curve √x, with largest gaps 1.000, 0.500, 0.259, 0.176, 0.108, 0.068.](/f/polynomials-climbing-to-the-square-root-each-above-the-last-ce601e49.svg)
Settling in order settles everywhere at once
A sequence of functions can settle at every point and never settle uniformly — the largest gap stays put while each point escapes it. Dini found the circumstances in which that cannot happen: continuous functions, a continuous limit, every member above the next, and a closed interval. Under those four conditions convergence at each point is convergence everywhere at once, and Weierstrass's iteration for the square root becomes a sequence of polynomials converging uniformly to |x|.
Close in height and nowhere close in slope
Uniform convergence carries continuity to the limit and carries integrals to the limit. It carries slopes nowhere. The functions sin(n²x)/n flatten to nothing while their slopes grow without bound; smooth curves converge uniformly to a corner; a Fourier series converges and its differentiated series diverges. What does carry a slope is uniform convergence of the slopes themselves — and the reason is that integration averages wiggles away while differentiation multiplies them.
Named alongside it
The objects these essays reach for when they reach for this one.
ContinuityCounterexampleUniform convergenceCompactnessPointwise convergenceApproximationArc lengthAxiom of choiceCardinalityCofinalityContinuum hypothesisConvergence