1.3 Axioms of Completeness
5 min read•july 30, 2024
The Axioms of Completeness are crucial in understanding the real number system. They ensure there are no gaps in the real number line, making it continuous and complete. This property sets real numbers apart from other number systems and is essential for calculus and mathematical analysis.
These axioms have far-reaching implications. They allow for the existence of irrational and transcendental numbers, enable powerful analytical tools, and are fundamental to defining integrals and proving key theorems in algebra and analysis.
Completeness in the Real Numbers
Fundamental Property and Significance
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- Completeness is a fundamental property of the real number system that distinguishes it from other number systems (rational numbers)
- The real number system is considered complete because it contains all the limit points of any Cauchy sequence of real numbers
- Completeness ensures that there are no "gaps" or "missing points" in the real number line, making it a continuous set
- The completeness property is crucial for the development of calculus and mathematical analysis guarantees the existence of limits, suprema, and infima
Implications and Applications
- Completeness allows for the existence of irrational numbers (√2, π) and transcendental numbers (e), which are not present in the rational number system
- The completeness property enables the use of powerful tools in analysis, such as the and the
- Completeness is essential for defining the Riemann integral, which is used to calculate areas, volumes, and other quantities in calculus
- The completeness of the real numbers is a necessary condition for the fundamental theorem of algebra, which states that every non-constant polynomial with complex coefficients has at least one complex root
Least Upper Bound Property
Definition and Equivalence to Completeness
- The , also known as the , states that every non-empty subset of real numbers that is bounded above has a least upper bound () in the set of real numbers
- A least upper bound (supremum) of a set A is the smallest real number that is greater than or equal to every element in A
- The Least Upper Bound Property is equivalent to the completeness of the real number system, as it ensures that there are no "gaps" in the real number line
- The existence of the least upper bound for any is a unique feature of the real number system and is not true for other number systems (rational numbers)
Importance and Applications
- The Least Upper Bound Property is used to prove the Archimedean Property, which states that for any positive real numbers x and y, there exists a natural number n such that nx > y
- The Least Upper Bound Property is essential for proving the existence of limits of sequences and functions, as well as the convergence of infinite series
- The Least Upper Bound Property is used to prove the Bolzano-Weierstrass Theorem, which states that every bounded sequence of real numbers has a convergent subsequence
- The Least Upper Bound Property is a key component in the construction of the Riemann integral, which relies on the existence of the supremum and of sets of real numbers
Proving the Least Upper Bound
Proof Outline
- To prove the existence of the least upper bound, consider a non-empty set A of real numbers that is bounded above
- Define a set B as the set of all upper bounds of A. By assumption, B is non-empty since A is bounded above
- Prove that B has a greatest lower bound (infimum) using the or the Nested Interval Property
- Show that the infimum of B is the least upper bound (supremum) of A by demonstrating that it is an upper bound of A and that no smaller number can be an upper bound of A
- Conclude that every non-empty set of real numbers that is bounded above has a least upper bound, thus proving the Least Upper Bound Property
Key Steps and Techniques
- Use the Axiom of Completeness, which states that every non-empty set of real numbers that is bounded above has a least upper bound, to prove the existence of the infimum of B
- Apply the Nested Interval Property, which states that if a sequence of closed, bounded intervals is nested (each interval contains the next), then the intersection of all the intervals is non-empty, to construct a sequence of nested intervals whose intersection contains the infimum of B
- Utilize the definition of the infimum to show that the infimum of B is an upper bound of A and that no smaller number can be an upper bound of A
- Apply proof by contradiction to demonstrate that the infimum of B is the least upper bound of A, by assuming that there exists a smaller upper bound and deriving a contradiction
- Use the properties of inequalities and the definition of the supremum to complete the proof
Completeness Axiom Applications
Convergence of Sequences and Series
- Use the to prove the convergence of monotonic and bounded sequences
- Apply the Monotone Convergence Theorem, which states that a monotonic sequence converges if and only if it is bounded, to solve problems involving the convergence of sequences
- Utilize the completeness axiom to prove the Cauchy Criterion for the convergence of sequences, which states that a sequence converges if and only if it is a Cauchy sequence
- Apply the completeness axiom to prove the convergence of series using the Cauchy Criterion for series or the Monotone Convergence Theorem for series
Continuity and Intermediate Value Theorem
- Use the completeness axiom to justify the existence of limits of functions and to prove the Intermediate Value Theorem, which relies on the continuity of the real number line
- The Intermediate Value Theorem states that if a function f is continuous on a closed interval [a, b] and k is a number between f(a) and f(b), then there exists a point c in [a, b] such that f(c) = k
- The completeness of the real numbers ensures that there are no "gaps" in the domain of a continuous function, allowing the Intermediate Value Theorem to hold
- Apply the Intermediate Value Theorem to solve problems involving the existence of roots of continuous functions on closed intervals
Key Terms to Review (13)
Axiom of Completeness: The axiom of completeness is a fundamental principle in real analysis that asserts every non-empty set of real numbers that is bounded above has a least upper bound (supremum). This axiom ensures that the real numbers are complete and provides a solid foundation for many important theorems and concepts, particularly in calculus and mathematical analysis. It distinguishes the real numbers from the rational numbers, which do not possess this property, leading to various implications in convergence and limits.
Bolzano-Weierstrass Theorem: The Bolzano-Weierstrass Theorem states that every bounded sequence in $b{R}^n$ has a convergent subsequence whose limit lies within the same space. This theorem is fundamental in understanding the properties of sequences and functions, particularly in the context of continuity and optimization.
Bounded set: A bounded set is a collection of elements in a given space that has both an upper and a lower limit, meaning that there exists a number that is greater than or equal to every element in the set and another number that is less than or equal to every element in the set. This concept is crucial for understanding the structure of real numbers, as it directly relates to the properties of least upper bounds and greatest lower bounds, ensuring that sets do not extend indefinitely in either direction.
Cauchy Sequences: Cauchy sequences are sequences of numbers where, for any small positive distance, there exists a point in the sequence beyond which all terms are within that distance from each other. This concept is crucial in understanding how sequences behave as they converge and is closely linked to completeness, as every Cauchy sequence converges in a complete space. The idea behind Cauchy sequences emphasizes that the terms of the sequence get arbitrarily close to each other as the sequence progresses, rather than necessarily approaching a specific limit.
Completeness axiom: The completeness axiom states that every non-empty set of real numbers that is bounded above has a least upper bound, or supremum. This principle ensures that the real numbers are 'complete' in the sense that there are no gaps, allowing for the definition and properties of limits, continuity, and convergence to be established clearly.
Dedekind Cut: A Dedekind cut is a method of constructing the real numbers from the rational numbers by partitioning the set of rational numbers into two non-empty subsets, A and B, where all elements of A are less than all elements of B. This concept is crucial for understanding how to address the completeness property of the real numbers, as it ensures that every real number corresponds to a unique cut, helping to eliminate gaps that exist within the rational numbers.
Heine-Borel Theorem: The Heine-Borel Theorem states that a subset of Euclidean space is compact if and only if it is closed and bounded. This theorem is essential in real analysis as it connects the concepts of compactness with two critical properties: closure and boundedness, making it easier to apply results related to continuity and convergence in the analysis of functions.
Infimum: The infimum, or greatest lower bound, of a set is the largest value that is less than or equal to every element in that set. This concept is critical in understanding limits and bounds of sequences and sets, particularly in the context of completeness, as it helps establish the existence of limits for monotone sequences and plays a key role in analyzing convergence.
Least upper bound property: 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) in the real numbers. This property is crucial because it ensures that there are no 'gaps' in the real number line, allowing for the precise definition of limits and continuity.
Limit Point: A limit point of a set is a point such that any neighborhood of this point contains at least one point from the set different from itself. Limit points play a critical role in understanding convergence, continuity, and the behavior of sequences, as they help define the limits and boundaries within mathematical analysis.
Order Topology: Order topology is a topology that arises from a totally ordered set, where the open sets are generated by intervals of the form $(-\infty, b)$ and $(a, \infty)$ for elements $a$ and $b$ in the set. This concept connects to the supremum and infimum by defining the structure of open sets that help establish limits and bounds within ordered sets. It also relates to the axioms of completeness, which ensure that every non-empty set of real numbers that is bounded above has a supremum, thereby reinforcing the foundational properties of order topology.
Supremum: The supremum, or least upper bound, of a set is the smallest number that is greater than or equal to every number in that set. This concept connects to various mathematical principles such as order structure and completeness, and it plays a crucial role in understanding limits, convergence, and the behavior of sequences.
Supremum property: The supremum property states that every non-empty set of real numbers that is bounded above has a least upper bound, or supremum. This means if you take any set of numbers that doesn’t go over a certain value, there’s always a highest point that is still less than or equal to all the numbers in that set. This property is essential for establishing the completeness of the real numbers and is crucial in defining limits, continuity, and integrals.