🏃🏽♀️➡️Intro to Mathematical Analysis Unit 4 – Completeness and Supremum/Infimum Properties
Completeness and supremum/infimum properties are fundamental concepts in real analysis. They provide a rigorous foundation for understanding the real number system and its unique characteristics compared to other number systems like rational numbers.
These properties are crucial for proving important theorems in calculus and analysis. They allow us to work with limits, continuity, and differentiability, forming the backbone of many advanced mathematical concepts and their applications in various fields.
Completeness a fundamental property of the real number system (R) ensures every non-empty subset of R that is bounded above has a least upper bound (supremum)
Supremum the smallest upper bound of a set A, denoted as supA, is the smallest real number greater than or equal to every element in A
Formally, s=supA if s is an upper bound of A and for any upper bound t of A, s≤t
Infimum the greatest lower bound of a set A, denoted as infA, is the largest real number less than or equal to every element in A
Formally, i=infA if i is a lower bound of A and for any lower bound t of A, i≥t
Upper bound a real number b is an upper bound of a set A if b≥a for all a∈A
Lower bound a real number b is a lower bound of a set A if b≤a for all a∈A
Bounded above a set A is bounded above if there exists a real number b such that a≤b for all a∈A
Bounded below a set A is bounded below if there exists a real number b such that a≥b for all a∈A
Historical Context and Importance
The concept of completeness emerged in the 19th century as mathematicians sought to rigorously define the real number system and its properties
Completeness plays a crucial role in establishing the foundations of real analysis, a branch of mathematics that deals with the properties of real numbers and functions of real variables
The completeness axiom distinguishes the real number system from other number systems, such as the rational numbers (Q), which are not complete
For example, the set of rational numbers between 0 and 2 whose squares are less than 2 does not have a rational supremum, highlighting the incompleteness of Q
Completeness is essential for proving the existence of limits, continuity, and differentiability of functions, which are fundamental concepts in real analysis
Many important theorems in real analysis, such as the Intermediate Value Theorem and the Extreme Value Theorem, rely on the completeness of the real number system
The completeness axiom also has practical implications in various fields, such as physics and engineering, where real numbers are used to model continuous quantities and phenomena
Completeness Axiom Explained
The completeness axiom states that every non-empty subset of real numbers that is bounded above has a least upper bound (supremum) in the set of real numbers
Formally, if A is a non-empty subset of R and A is bounded above, then there exists a real number s such that s=supA
The completeness axiom is an essential property that distinguishes the real number system from other number systems, such as the rational numbers
One consequence of the completeness axiom is the Archimedean property, which states that for any positive real numbers a and b, there exists a natural number n such that na>b
This property ensures that there are no "infinitely large" or "infinitely small" real numbers
The completeness axiom is closely related to the notion of Dedekind cuts, which provide a way to construct the real numbers from the rational numbers
A Dedekind cut is a partition of the rational numbers into two non-empty sets A and B, such that every element of A is less than every element of B, and A has no greatest element
The completeness axiom ensures that every Dedekind cut corresponds to a unique real number
Supremum and Infimum: The Basics
The supremum (sup) of a set A is the least upper bound of A, denoted as supA
If s=supA, then s is an upper bound of A, and for any upper bound t of A, s≤t
The infimum (inf) of a set A is the greatest lower bound of A, denoted as infA
If i=infA, then i is a lower bound of A, and for any lower bound t of A, i≥t
A set A may have a supremum or infimum that is not an element of A itself
For example, the set A=(0,1) has supA=1 and infA=0, neither of which are elements of A
If a set A has a maximum element m, then supA=m; similarly, if A has a minimum element n, then infA=n
The supremum and infimum of a set A are unique when they exist, as a consequence of the completeness axiom
For any non-empty set A that is bounded above, supA always exists in R; similarly, for any non-empty set A that is bounded below, infA always exists in R
Properties and Theorems
The Supremum Property if A is a non-empty subset of R that is bounded above, then supA exists in R
The Infimum Property if A is a non-empty subset of R that is bounded below, then infA exists in R
Theorem if A and B are non-empty subsets of R with A⊆B, then supA≤supB and infA≥infB
Theorem for any real number c and non-empty set A, sup(c+A)=c+supA and inf(c+A)=c+infA, where c+A={c+a:a∈A}
Theorem for any positive real number c and non-empty set A, sup(cA)=c⋅supA and inf(cA)=c⋅infA, where cA={ca:a∈A}
If c is negative, then sup(cA)=c⋅infA and inf(cA)=c⋅supA
The Nested Interval Property if {In} is a sequence of closed, bounded, and nested intervals (i.e., In+1⊆In for all n), then the intersection ⋂n=1∞In is non-empty
This property is a consequence of the completeness axiom and is crucial for proving the existence of certain limits
Applications in Real Analysis
The completeness axiom is essential for proving the existence of limits of sequences and functions
For example, the Monotone Convergence Theorem states that if a sequence is monotone (either non-decreasing or non-increasing) and bounded, then it converges to a limit in R
Completeness is crucial for establishing the continuity and differentiability of functions
The Intermediate Value Theorem, which relies on completeness, states that if f is a continuous function on [a,b] and y is between f(a) and f(b), then there exists a c∈(a,b) such that f(c)=y
The Extreme Value Theorem, another consequence of completeness, states that a continuous function on a closed and bounded interval attains its maximum and minimum values
Completeness is essential for defining the Riemann integral, which is used to calculate the area under a curve
The Riemann integral of a function f on [a,b] is defined as the limit of Riemann sums, which rely on the completeness of R
The Heine-Borel Theorem, which states that a subset of Rn is compact if and only if it is closed and bounded, is a consequence of the completeness axiom
This theorem has important applications in topology and functional analysis
Common Pitfalls and Misconceptions
Confusing completeness with other properties, such as density or continuity
Density means that between any two real numbers, there exists another real number, which is true for both Q and R
Continuity is a property of functions, not number systems
Assuming that every set has a supremum or infimum
Sets that are not bounded above or below may not have a supremum or infimum (e.g., R itself)
Empty sets do not have a supremum or infimum
Thinking that the supremum or infimum of a set must be an element of the set
The supremum or infimum may not be contained in the set itself (e.g., sup(0,1)=1∈/(0,1))
Misunderstanding the relationship between supremum, maximum, infimum, and minimum
A set may have a supremum or infimum without having a maximum or minimum element
If a set has a maximum or minimum element, it will be equal to the supremum or infimum, respectively
Incorrectly applying properties of supremum and infimum
When multiplying a set by a negative number, the supremum and infimum are swapped (i.e., sup(cA)=c⋅infA and inf(cA)=c⋅supA for c<0)
Forgetting that the completeness axiom is specific to the real number system
Other number systems, such as the rational numbers or the hyperreal numbers, may not satisfy the completeness axiom
Practice Problems and Solutions
Find the supremum and infimum of the set A={x∈R:x2<5}.
Solution supA=5 and infA=−5. The set A consists of all real numbers whose squares are less than 5. The largest such number is 5, and the smallest is −5. Note that 5∈/A and −5∈/A.
Prove that if A and B are non-empty subsets of R with A⊆B, then supA≤supB.
Solution Let sA=supA and sB=supB. Since A⊆B, every element of A is also an element of B. Therefore, sB is an upper bound for A. By the definition of supremum, sA is the least upper bound of A, so sA≤sB.
Let A={1−n1:n∈N}. Find supA and infA.
Solution supA=1 and infA=0. The sequence {1−n1} is increasing and bounded above by 1, so its supremum is 1. The infimum is 0 because 1−n1>0 for all n∈N, and as n approaches infinity, 1−n1 gets arbitrarily close to 0.
Prove that if a<b, then there exists a rational number r such that a<r<b.
Solution Let a,b∈R with a<b. Consider the set A={r∈Q:r≤a}. Since a is an upper bound for A and Q is dense in R, there exists an r∈Q such that a<r<b.
Prove that the set {x∈Q:x2<2} does not have a supremum in Q.
Solution Suppose, for the sake of contradiction, that the set A={x∈Q:x2<2} has a supremum s in Q. Then s2≤2. If s2=2, then s=2∈/Q, a contradiction. If s2<2, then there exists a rational number r such that s<r<2, which means r∈A and r>s, contradicting the fact that s is the supremum of A. Therefore, A does not have a supremum in Q.