The Least Upper Bound Property is a key concept in real analysis. It states that every non-empty set of real numbers with an upper bound has a least upper bound. This property distinguishes real numbers from other number systems and ensures there are no gaps in the real number line.
Understanding this property is crucial for grasping completeness in real numbers. It's used to prove important theorems like Bolzano-Weierstrass and the Extreme Value Theorem. The Least Upper Bound Property is fundamental in constructing real numbers and forms the basis for many advanced concepts in analysis.
Least Upper Bound Property
Definition and Terminology
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The least upper bound property states that every non-empty set of real numbers that is bounded above has a least upper bound (or supremum)
A real number b is called an upper bound for a set S if x≤b for all x in S
A real number b is the least upper bound (or supremum) of a set S if b is an upper bound for S and b≤y for every upper bound y of S
For example, consider the set S={x∈R:0<x<1}. The number 1 is the least upper bound of S because it is an upper bound and is less than or equal to any other upper bound of S
The greatest lower bound (or infimum) of a set S is defined similarly, with the inequality reversed
For example, in the set S={x∈R:0<x<1}, the number 0 is the greatest lower bound of S because it is a lower bound and is greater than or equal to any other lower bound of S
The least upper bound property is also known as the supremum property or the completeness property
Significance in Real Analysis
The least upper bound property is a fundamental concept in real analysis and is one of the key properties that distinguish the real numbers from other number systems (rational numbers)
It ensures that the real number system has no "gaps" or "holes" and that every bounded set of real numbers has a well-defined supremum and infimum
The least upper bound property is essential for proving many important theorems in real analysis, such as the Bolzano-Weierstrass theorem, the Heine-Borel theorem, and the Extreme Value Theorem
It also plays a crucial role in the construction of the real numbers from the rational numbers using Dedekind cuts or Cauchy sequences
Proving the Least Upper Bound Property
Dedekind Cut Method
The Dedekind cut method defines a real number as a partition of the set of rational numbers into two non-empty sets A and B, such that every element of A is less than every element of B
A Dedekind cut (A,B) is said to represent a real number α if A contains all rational numbers less than α and B contains all rational numbers greater than or equal to α
The least upper bound property can be proved using Dedekind cuts by showing that for any non-empty, bounded above set of real numbers S, there exists a Dedekind cut (A,B) such that A is the set of all rational numbers less than or equal to any element of S, and B is the set of all rational numbers greater than any element of S
This Dedekind cut represents the least upper bound of S, proving the least upper bound property for the real numbers
Cauchy Sequence Method
The Cauchy sequence method defines a real number as the limit of a Cauchy sequence of rational numbers
A sequence (an) of rational numbers is called a Cauchy sequence if for every ε>0, there exists an N∈N such that ∣am−an∣<ε for all m,n≥N
Two Cauchy sequences (an) and (bn) are said to be equivalent if limn→∞(an−bn)=0. The set of equivalence classes of Cauchy sequences forms the real numbers
The least upper bound property can be proved using Cauchy sequences by showing that for any non-empty, bounded above set of real numbers S, there exists a Cauchy sequence (an) such that limn→∞an is the least upper bound of S
This proves the least upper bound property for the real numbers constructed using Cauchy sequences
Subsets with Least Upper Bounds
Bounded Sets
Bounded sets of real numbers always have a least upper bound and a greatest lower bound
A set S is bounded above if there exists a real number M such that x≤M for all x∈S. Similarly, S is bounded below if there exists a real number m such that m≤x for all x∈S
If a set is both bounded above and bounded below, it is called a bounded set
For example, the set S={x∈R:0≤x≤1} is bounded, with a least upper bound of 1 and a greatest lower bound of 0
Intervals
Closed intervals [a,b] and half-closed intervals [a,b) and (a,b] have a least upper bound
For example, the interval [0,1] has a least upper bound of 1, and the interval [0,1) has a least upper bound of 1
Open intervals (a,b) do not have a least upper bound or greatest lower bound, as they do not contain their endpoints
For example, the open interval (0,1) does not have a least upper bound or greatest lower bound, as any number less than 1 is not an upper bound, and any number greater than 0 is not a lower bound
Unbounded sets, such as the set of all real numbers R or the set of positive real numbers R+, do not have a least upper bound
Applying the Least Upper Bound Property
Proving Existence of Limits, Suprema, and Infima
The least upper bound property is used to prove the existence of limits, suprema, and infima in real analysis
For example, to prove that a sequence (an) converges to a limit L, we can show that the set {an:n∈N} is bounded and that L is the least upper bound (or greatest lower bound) of the set {an:n≥N} for sufficiently large N
Similarly, to prove that a function f attains its supremum or infimum on a closed interval [a,b], we can use the least upper bound property to show that the set {f(x):x∈[a,b]} has a least upper bound (or greatest lower bound) that is attained by f at some point in [a,b]
Important Theorems
The Bolzano-Weierstrass theorem, which states that every bounded sequence of real numbers has a convergent subsequence, relies on the least upper bound property
The proof involves constructing a nested sequence of closed intervals using the least upper bound property and showing that their intersection contains a single point, which is the limit of a subsequence
The least upper bound property is used to prove the Extreme Value Theorem, which states that a continuous function on a closed interval attains its maximum and minimum values
The proof involves showing that the set of function values is bounded and has a least upper bound and greatest lower bound, which are attained by the function due to its continuity
The Heine-Borel theorem, which characterizes compact subsets of Euclidean space, is proved using the least upper bound property
The proof involves showing that a set is compact if and only if it is closed and bounded, using the least upper bound property to construct finite subcovers from open covers
The least upper bound property is used to prove the existence of the Riemann integral for bounded functions on closed intervals
The proof involves constructing upper and lower Riemann sums using the least upper bound and greatest lower bound of the function on subintervals and showing that their difference can be made arbitrarily small