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

Properties of the Absolute Value 3 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

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

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

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

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

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

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 1 2 1 and 2 Mika Seppälä: Sup and Inf