1. Supremum and Infimum
Mika Seppälä

2. 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).
Definition
The distance between two real numbers x and y is |x-y|.
Mika Seppälä: Sup and Inf

3. Properties of the Absolute Value3 1 2 4 5 6 6 7
Triangle Inequality
Example
Proof
Problem
When do we have equality in the above estimate?
Mika Seppälä: Sup and Inf

4. Solving Absolute Value Equations
Example
Solution
Conclusion
The equation has two solutions: x = 2 and x = -3.
Mika Seppälä: Sup and Inf

5. Solving Absolute Value Inequalities
Example
Solution
Conclusion
Mika Seppälä: Sup and Inf

6. 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

7. 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.
Notation
sup(A) = the supremum of the set A.
Example
Mika Seppälä: Sup and Inf

8. 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.
Notation
inf(A) = the infimum of the set A.
Example
Mika Seppälä: Sup and Inf

9. Characterization of the Supremum (1)
Theorem
Proof
Mika Seppälä: Sup and Inf

10. Characterization of the Supremum (2)
Theorem
Proof
Cont’d
Mika Seppälä: Sup and Inf

11. 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

12. Usage of the Characterizations
Example
Claim
Proof of the Claim 1 2 1
and 2
Mika Seppälä: Sup and Inf