Suprema, Infima, and the Completeness Property

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Definition of Supremum and Infimum

Definition: Let be a set that is bounded above. We say that the supremum of denoted is a number that satisfies the conditions that is an upper bound of and is the least upper bound of , that is for any that is also an upper bound of then .

Definition: Let be a set that is bounded below. We say that the infimum of denoted is a number that satisfies the conditions that is a lower bound of and is the greatest lower bound of , that is for any that is also a lower bound of then .

We will now reformulate these definitions with an equivalent statement that may be useful to apply in certain situations in showing that an upper bound is the supremum of a set, or showing that a lower bound is the infimum of a set.

Theorem 1: Let be a nonempty subset of the real numbers that is bounded above. The upper bound is said to be the supremum of if and only if there exists an element such that .

Supremum of set
  • Proof: Let be a nonempty subset of the real numbers that is bounded above. We first want to show that if is an upper bound such that there exists an element such that , then . Let be an upper bound of that satisfies the condition stated above, and suppose that . Then choose , and so and so there exists an element such that , and thus, .
  • We now want to show that if then there exists an element such that . Let and let . We note that and so is not an upper bound of the set . Therefore by the definition that , there exists some element such that .

Theorem 2: Let be a nonempty subset of the real numbers that is bounded below. The lower bound is said to be the infimum of if and only if there exists an element such that .

Infimum of set
  • Proof: Let be a nonempty subset of the real numbers that is bounded below. We first want to show that if is a lower bound such that there exists an element such that , then . Let be a lower bound of that satisfies the condition stated above, and suppose that . Then choose , and so and so there exists an element such that , and thus, .
  • We now want to show that if then there exists an element such that . Let and let . We note that and so is not a lower bound of the set . Therefore by the definition that , there exists some element such that .

The Supremum and Infimum of a Function

We will now begin to look at some applications of the definition of a supremum and infimum with regards to functions.

Definition: Let be a function. Then define the supremum of to be , and define the infimum of to be where is the range of .

From the definition above, we acknowledge that the supremum and infimum of a function pertain to the set that is the range of . The diagram below illustrates the supremum and infimum of a function:

Supremum and infimum of function

We will now look at some important theorems.

Theorem 1: Let and be functions such that is bounded above. If for all , then .

  • Proof: Let . Let be a function that is bounded above. Since the range of is nonempty we have that for all , . Now since for all , we have that .
  • Furthermore we note that since for all , then is bounded below by , and so and thus .

Theorem 2: Let and be functions such that is bounded below. If for all , then .

  • Proof: Let be a function that is bounded below. Let . Since the range of is nonempty we have that for all , . Now since for all we have that .
  • Furthermore we note that since for all , then is bounded above by , and so and thus .

Theorem 3: Let be a bounded function and let . If then , and . If then , and .

Theorem 3 immediately follows from the theorems we've already proven on <a href="/the-supremum-and-infimum-of-the-bounded-set-as">The Supremum and Infimum of The Bounded Set (aS)</a> page where .

The Completeness Property of The Real Numbers

We will now look at yet again another crucially important property of the real numbers which will allow us to call the set of numbers under the operations of addition and multiplication a complete ordered field. This property will ensure that there is no "gaps" in the real number line, that is the real number line is continuous. The property is as follows.

The Completeness Property of The Real Numbers: Every nonempty subset of the real numbers that is bounded above has a supremum in .

The Completeness Property is also often called the "Least Upper Bound Property".

The completeness property above is a crucial axiom. A similar theorem regarding nonempty subsets of the real numbers that are bounded below exists and is proven below.

Theorem 1: Every nonempty subset of the real numbers that is bounded below has an infimum in .

  • Proof: Let be a nonempty subset of real numbers that is bounded below. Then there exists a lower bound such that for every .
  • Define the set by:
  • Then observe that for every . So is a nonempty subset of real numbers that is bounded above. By the completeness property of the real numbers we have that has a supremum in . Let . Then for every and if is any other upper bound of then .
  • So observe that for every and if is any other lower bound of then . So . In other words, has an infimum in .

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