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4.2Estimated reading time: 6 min

4.2 Upper bounds, supremum, and infimum

Distinguish maxima and minima from upper and lower bounds, then learn how supremum and infimum capture the correct extremal language.

Course contents

MATH1090: Set theory

Rigorous course notes on logic, sets, number construction, the real numbers, limits, cardinality, and the first algebraic structures, written in linked sections with careful proofs and examples.

Language

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All note collectionsSeries overview
1234567
Chapter 1Logic3 sections
1.11.1 Propositional logic1.21.2 Truth tables and equivalence1.31.3 Quantifiers and negation
Chapter 2Sets and relations2 sections
2.12.1 Sets and set operations2.22.2 Functions and relations
Chapter 3Numbers by construction5 sections
3.13.1 Natural numbers and Peano axioms3.23.2 Induction and recursive arithmetic3.33.3 Integers from equivalence classes3.43.4 Rationals and well-defined operations3.53.5 Gaps in Q and why sqrt(2) is not rational
Chapter 4Order and completeness6 sections
4.14.1 Total orders and ordered fields4.24.2 Upper bounds, supremum, and infimumCurrent section4.34.3 Completeness and gaps in Q4.44.4 Axioms for the reals and first approximations4.54.5 Dedekind cuts and the embedding of Q4.64.6 Decimal expansions and irrational numbers
Chapter 5Sequences and first limits3 sections
5.15.1 Sequences and epsilon-N limits5.25.2 Cauchy sequences and another model of the reals5.35.3 Delta-epsilon limits, limit laws, and continuity
Chapter 6Big sets3 sections
6.16.1 Cardinality, countability, and cardinal inequalities6.26.2 Cantor's theorem, continuum, and choice6.36.3 Intervals, Cantor set, density, and well-ordering
Chapter 7Sets with structure1 sections
7.17.1 Binary operations, monoids, and groups

Why we need precise language for extremes

Once a set is ordered, the interesting questions usually happen near its extremes. Does the set have a largest element? If not, is there still a smallest number sitting above all of it? Is there a best lower barrier from the left?

The key chain of ideas is:

  • maximum and minimum talk about elements that lie inside the set;
  • upper and lower bounds may lie outside the set;
  • supremum and infimum identify the best possible such bounds.

If you mix these ideas together, later arguments about completeness become hard to read. So this note separates them carefully.

Maximum and minimum versus upper and lower bounds

Definition

Maximum, minimum, upper bound, and lower bound

Let YY be a subset of an ordered set XX.

  • A maximum of YY is an element m∈Ym\in Y such that y≤my\le m for every y∈Yy\in Y.
  • A minimum of YY is an element n∈Yn\in Y such that n≤yn\le y for every y∈Yy\in Y.
  • An upper bound of YY is an element u∈Xu\in X such that y≤uy\le u for every y∈Yy\in Y.
  • A lower bound of YY is an element ℓ∈X\ell\in X such that ℓ≤y\ell\le y for every y∈Yy\in Y.

The difference is subtle but essential: a maximum or minimum must belong to the set, while an upper or lower bound only has to belong to the ambient ordered set.

Worked example

A finite set in Z

Let Y={1,2,3}⊆ZY=\{1,2,3\}\subseteq Z.

  • The maximum is 3.
  • The minimum is 1.
  • Every integer u≥3u\ge 3 is an upper bound.
  • Every integer ℓ≤1\ell\le 1 is a lower bound.

So the set has many upper and lower bounds, but only one maximum and one minimum.

Worked example

A set can have bounds without extremal elements

Consider Q>0Q_{>0} as a subset of QQ.

The number 0 is a lower bound of Q>0Q_{>0}, but it is not an element of the set. In fact, Q>0Q_{>0} has no minimum, because for every positive rational q, the smaller rational q/2 is still positive.

This example is a warning: lower bound and minimum are not interchangeable.

Supremum and infimum

Some upper bounds are better than others. The supremum is the smallest upper bound; the infimum is the largest lower bound.

Definition

Supremum and infimum

Let YY be a nonempty subset of an ordered set XX.

  • A number s∈Xs\in X is the supremum of YY, written s=sup⁡(Y)s=\sup(Y), if s is an upper bound of YY and every upper bound u of YY satisfies s≤us\le u.
  • A number t∈Xt\in X is the infimum of YY, written t=inf⁡(Y)t=\inf(Y), if t is a lower bound of YY and every lower bound ℓ\ell of YY satisfies ℓ≤t\ell\le t.

The ambient set matters. The same subset can have a supremum in RR but fail to have one in QQ.

Worked example

The open interval (0,1)

Take Y=(0,1)⊆RY=(0,1)\subseteq R.

  • YY has no maximum, because 1 is not in the set and every element is still smaller than some larger element of YY.
  • sup⁡(Y)=1\sup(Y)=1.
  • YY has no minimum.
  • inf⁡(Y)=0\inf(Y)=0.

So both extremal bounds exist even though neither belongs to the set.

Why supremum and infimum are unique

Theorem

A subset has at most one supremum and at most one infimum

If Y⊆XY\subseteq X has a supremum, then that supremum is unique. The same holds for infimum.

The reason is short but important. Suppose s and s' are both suprema of YY. Since s is an upper bound and s' is the least upper bound, we get s′≤ss'\le s. Reversing the roles gives s≤s′s\le s'. By antisymmetry, s=s′s=s'.

Exactly the same argument works for infimum.

The approximation viewpoint

The definition of supremum can be turned into a more usable test.

s=sup⁡(Y)s=\sup(Y) if and only if s is an upper bound of YY and for every ε>0\varepsilon>0 there exists y∈Yy\in Y such that s−ε<y≤ss-\varepsilon<y\le s.

This says that a supremum is not just an upper wall; points of the set can come arbitrarily close to it from below.

Proof

Why the approximation property is equivalent

Common mistakes

Common mistake

Maximum is stronger than supremum

A maximum must belong to the set. A supremum only has to be the least upper bound in the ambient ordered set. The open interval (0,1) has supremum 1 but no maximum.

Common mistake

The ambient ordered set matters

When you write sup⁡(Y)\sup(Y), you are working inside some ordered set. A subset of QQ can have a supremum in RR without having one in QQ. Later, this is exactly how we see that QQ is not complete.

Quick checks

Quick check

For Y=(0,1), does Y have a maximum? What are sup(Y) and inf(Y)?

Keep the inside/outside distinction clear.

Solution

Answer

Quick check

If a set A has a maximum m, what is sup(A)?

Compare the definitions directly.

Solution

Answer

Exercises

Quick check

Let A={1-1/n : n in N}. Find sup(A), inf(A), and decide whether A has a maximum.

List the first few terms and look at the trend.

Solution

Guided solution

Quick check

Show that if B is nonempty and bounded below, then inf(B) = -sup(-B).

Translate lower-bound statements for B into upper-bound statements for -B.

Solution

Guided solution

Related notes

Read this after 4.1 Total orders and ordered fields and continue to 4.3 Completeness and gaps in Q.

Prerequisites

4.1

4.1 Total orders and ordered fields

Key terms in this unit

More notes in this series

Previous section

4.1 Total orders and ordered fields

Next section

4.3 Completeness and gaps in Q