Glossary

and are key concepts in real analysis, helping us understand the boundaries of sets. They're like the ultimate upper and lower limits, even for sets without clear max or min values.

These ideas are crucial for completeness in real numbers. By grasping supremum and infimum, we can better understand how sets behave and why the real number system is so powerful in mathematics.

Supremum and Infimum of Sets

Definition and Notation

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  • The supremum of a set S of real numbers is the of S, denoted as (S)
    • It is the smallest real number that is greater than or equal to every element in S
    • Example: For the set S = {1, 2, 3, 4}, sup(S) = 4
  • The infimum of a set S of real numbers is the of S, denoted as (S)
    • It is the largest real number that is less than or equal to every element in S
    • Example: For the set S = {1, 2, 3, 4}, inf(S) = 1

Relationship with Maximum and Minimum

  • The supremum and infimum of a set S may or may not be elements of the set S itself
  • If the supremum (or infimum) of a set S is an element of S, then it is also the maximum (or minimum) of the set
    • Example: For the set S = {1, 2, 3, 4}, sup(S) = max(S) = 4 and inf(S) = min(S) = 1
  • The supremum and infimum of a set are unique when they exist
    • If a set has a supremum or infimum, there cannot be another distinct value that satisfies the definition of supremum or infimum for that set

Existence of Supremum and Infimum

Bounded Sets

  • A set S of real numbers is if there exists a real number M such that x ≤ M for all x in S
    • Example: The set S = {1, 2, 3, 4} is bounded above by M = 5
  • A set S is if there exists a real number m such that m ≤ x for all x in S
    • Example: The set S = {1, 2, 3, 4} is bounded below by m = 0
  • A set S is bounded if it is both bounded above and bounded below
  • The supremum and infimum always exist for non-empty bounded sets of real numbers

Unbounded Sets

  • For unbounded sets, the supremum (or infimum) may not exist
    • If a set is not bounded above, then it does not have a supremum
      • Example: The set of natural numbers N = {1, 2, 3, ...} is not bounded above and has no supremum
    • If a set is not bounded below, it does not have an infimum
      • Example: The set of negative real numbers {x ∈ R | x < 0} is not bounded below and has no infimum
  • The set of all real numbers, R, is unbounded and has neither a supremum nor an infimum

Properties of Supremum and Infimum

Uniqueness

  • If a set S has a supremum (or infimum), then it is unique
    • Proof by contradiction: Assume there exist two distinct suprema (or infima) for a set S, and show that this leads to a contradiction
    • Let sup1(S) and sup2(S) be two distinct suprema for the set S
    • Since sup1(S) is the supremum, sup1(S) ≤ sup2(S), and since sup2(S) is the supremum, sup2(S) ≤ sup1(S)
    • This implies sup1(S) = sup2(S), contradicting the assumption that they are distinct

Relationship with Upper and Lower Bounds

  • If a set S has a supremum sup(S), then sup(S) is an upper bound of S, and for any upper bound M of S, sup(S) ≤ M
    • Example: For the set S = {1, 2, 3, 4}, sup(S) = 4 is an upper bound, and for any upper bound M ≥ 4, sup(S) ≤ M
  • If a set S has an infimum inf(S), then inf(S) is a lower bound of S, and for any lower bound m of S, m ≤ inf(S)
    • Example: For the set S = {1, 2, 3, 4}, inf(S) = 1 is a lower bound, and for any lower bound m ≤ 1, m ≤ inf(S)
  • If a set S has a maximum (or minimum) element, then the maximum (or minimum) is equal to the supremum (or infimum) of the set

Calculating Supremum and Infimum

Determining Boundedness

  • To find the supremum (or infimum) of a set S, first determine if the set is bounded above (or below)
  • If the set is bounded, consider the properties of the set and its elements to identify the least upper bound (or greatest lower bound)
  • If the set is unbounded above (or below), then the supremum (or infimum) does not exist

Sets Defined by Intervals

  • For an open interval (a, b), sup(S) = b and inf(S) = a
    • Example: For the set S = (1, 5), sup(S) = 5 and inf(S) = 1
  • For a closed interval [a, b], sup(S) = max(S) = b and inf(S) = min(S) = a
    • Example: For the set S = [1, 5], sup(S) = max(S) = 5 and inf(S) = min(S) = 1
  • For a half-open interval (a, b] or [a, b), sup(S) = b and inf(S) = a
    • Example: For the set S = (1, 5] or S = [1, 5), sup(S) = 5 and inf(S) = 1

Sets Defined by Inequalities or Other Conditions

  • For sets defined by inequalities or other conditions, analyze the properties of the set to determine the supremum and infimum
  • Example: For the set S = {x ∈ R | 0 < x < 1}, sup(S) = 1 and inf(S) = 0
  • Example: For the set S = {1/n | n ∈ N}, sup(S) = 1 and inf(S) = 0

Key Terms to Review (18)

Bounded above: A set of numbers is said to be bounded above if there exists a real number that is greater than or equal to every number in the set. This means that all elements of the set do not exceed a certain limit, making it essential in understanding the behavior of sequences and functions. Boundedness leads to the concepts of supremum and infimum, which help in identifying the least upper bound of a set and understanding the properties of real numbers, intervals, and their absolute values.
Bounded below: A set is considered bounded below if there exists a real number that serves as a lower limit for the elements in that set. This means that no element in the set is less than this lower limit, providing a boundary that the elements cannot fall below. Understanding this concept is crucial for grasping related ideas like supremum and infimum, as well as recognizing the significance of the greatest lower bound property, which states that every non-empty set of real numbers that is bounded below has a greatest lower bound or infimum.
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.
Functional Analysis: Functional analysis is a branch of mathematical analysis that deals with the study of vector spaces and the linear operators acting upon them. It extends concepts from linear algebra and calculus to infinite-dimensional spaces, providing a framework for understanding the behavior of functions and their properties in various contexts. This discipline is crucial for analyzing convergence, completeness, and the interplay between various mathematical structures, connecting deeply with supremum and infimum concepts, completeness of sequences, and convergence types.
Greatest Lower Bound: The greatest lower bound, also known as the infimum, of a set is the largest value that is less than or equal to every element in that set. This concept is essential in understanding how sets behave within the real numbers, particularly when discussing bounded sets and the completeness property of the real numbers. The greatest lower bound plays a critical role in defining limits, continuity, and convergence within mathematical analysis.
Inf: The term 'inf', short for infimum, refers to the greatest lower bound of a set in mathematics. It is the largest value that is less than or equal to every element in the set, providing crucial insights into the behavior and properties of sets, particularly in analysis. Understanding infimum helps in establishing limits and bounds, which are essential for various mathematical concepts, including convergence and continuity.
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: The least upper bound, or supremum, of a set is the smallest number that is greater than or equal to every element in that set. This concept is crucial because it helps in understanding the bounds of a set and plays a key role in establishing whether a set has a maximum value or not. The least upper bound provides insights into the structure of real numbers and is closely tied to ideas such as convergence and limits, which are fundamental in mathematical analysis.
Limit Inferior: The limit inferior of a sequence is the greatest lower bound of the set of its subsequential limits. It gives a way to analyze the long-term behavior of a sequence, particularly when it fluctuates, and is closely related to concepts such as supremum and infimum in order to understand the bounds of sequences. This concept helps identify the lowest value that a sequence can approach infinitely often, reflecting its behavior at infinity.
Limit Superior: The limit superior of a sequence is the largest limit point of that sequence, representing the supremum of the set of its limit points. This concept captures the behavior of a sequence as it approaches infinity, helping to identify the upper bounds of its oscillations. By understanding limit superior, one can analyze sequences that do not converge in a traditional sense and can connect this idea to notions like supremum and infimum.
Measure theory: Measure theory is a branch of mathematical analysis that deals with the quantification of size or volume of sets, extending the concept of length, area, and volume in a rigorous way. It forms the foundation for probability, integration, and functional analysis, helping to define notions like convergence and continuity in a more generalized framework. This theory provides tools for working with infinite sets and measurable functions, making it essential for understanding limits and bounds in analysis.
Monotone Convergence Theorem: The Monotone Convergence Theorem states that if a sequence of real numbers is monotonic (either non-decreasing or non-increasing) and bounded, then it converges to a limit. This theorem is crucial as it connects the behavior of sequences with completeness and provides insights into the concepts of supremum and infimum.
Open Intervals: An open interval is a set of real numbers that includes all numbers between two given endpoints, but does not include the endpoints themselves. This concept is crucial when discussing limits, continuity, and the definitions of supremum and infimum, as it helps to understand where values can exist without being restricted by boundary points.
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.
Ordered Set: An ordered set is a collection of elements arranged in a specific sequence, where the order of elements is significant. This means that for any two elements in the set, one can be compared to another to determine their relative position or rank. Ordered sets provide a framework for discussing concepts such as supremum and infimum, where understanding the arrangement of elements is crucial for determining bounds and limits.
Set of Rational Numbers: The set of rational numbers consists of all numbers that can be expressed as the quotient of two integers, where the denominator is not zero. This set includes integers, finite decimals, and repeating decimals, making it a dense subset of the real numbers. The rational numbers are significant in analysis due to their properties in limits, supremum, and infimum calculations.
Sup: The term 'sup' stands for supremum, which is the least upper bound of a set of real numbers. This means it is the smallest number that is greater than or equal to every number in that set. Understanding the concept of supremum is crucial, especially when dealing with bounds of sequences and sets, as it helps in analyzing limits and convergence behavior.
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.
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