Supremum and Infimum

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Supremum and Infimum. Mika Seppälä. Distance in the Set of Real Numbers. Definition. Triangle Inequality.

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Supremum and Infimum

Mika Seppälä

Distance in the Set of Real Numbers

Definition

Triangle Inequality

Triangle inequality for the absolute value is almost obvious. We have equality on the right hand side if x and y are either both positive or both negative (or one of them is 0).

We have equality on the left hand side if the signs of x and y are opposite (or if one of them is 0).

The distance between two real numbers x and y is |x-y|.

Definition

Mika Seppälä: Sup and Inf

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Triangle Inequality

7

Properties of the Absolute Value

Example

Proof

Problem

When do we have equality in the above estimate?

Mika Seppälä: Sup and Inf

Solving Absolute Value Equations

Example

Solution

The equation has two solutions: x = 2 and x = -3.

Conclusion

Mika Seppälä: Sup and Inf

Solving Absolute Value Inequalities

Example

Solution

Conclusion

Mika Seppälä: Sup and Inf

Upper and Lower Bounds

Definition

Let A be a non-empty set of real numbers.

A set A need not have neither upper nor lower bounds.

The set A is bounded from above if A has a finite upper bound.

The set A is bounded from below if A has a finite lower bound.

The set A is bounded if it has finite upper and lower bounds.

Mika Seppälä: Sup and Inf

Supremum

Completeness of Real Numbers

The set A has finite upper bounds. An important completeness property of the set of real numbers is that the set A has a unique smallest upper bound.

Definition

The smallest upper bound of the set A is called the supremum of the set A.

sup(A) = the supremum of the set A.

Notation

Example

Mika Seppälä: Sup and Inf

Infimum

The set A has finite lower bounds. As in the case of upper bounds, the set of real numbers is complete in the sense that the set A has a unique largest lower bound.

Definition

The largest lower bound of the set A is called the infimum of the set A.

inf(A) = the infimum of the set A.

Notation

Example

Mika Seppälä: Sup and Inf

Characterization of the Supremum (1)

Theorem

Proof

Mika Seppälä: Sup and Inf

Characterization of the Supremum (2)

Theorem

Proof

Cont’d

Mika Seppälä: Sup and Inf

Characterization of the Infimum

Theorem

The proof of this result is a repetition of the argument the previous proof for the supremum.

Mika Seppälä: Sup and Inf

Usage of the Characterizations

Example

Claim

Proof of the Claim

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and

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Mika Seppälä: Sup and Inf