Supremum and infimum 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

• The supremum of a set S of real numbers is the least upper bound of S, denoted as sup(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 greatest lower bound of S, denoted as inf(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 bounded above 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 bounded below 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