1. Calculus and Elementary Analysis 1.2 Supremum and Infimum Integers, Rational Numbers and Real Numbers Completeness of Real Numbers Supremum and Infimum Characterizations of Sup and of Inf

2. 2015-5-12 Calculus and Elementary Analysis 1.2.1 Integers, Rational and Real Numbers Integers Notations Natural Numbers Rational Numbers This condition means that p and q have no common factors. Theorem There are no rational numbers r such that r 2 =2. Real Numbers Real numbers consist of rational and of irrational numbers. Notation

3. 2015-5-12 Calculus and Elementary Analysis Integers, Rational and Real Numbers Assume the contrary. This means that p 2 =2q 2. Proposition There are not rational numbers r such that r 2 =2. Proof Then there are positive integers p and q such that p 2 /q 2 =2 and p and q do not have common factors. Hence the area of a square with side length p is twice the area of the square with side length q. Furthermore p and q are smallest such numbers since they do have common factors. Graphically: Now p and q are integers so that B=p 2 =2q 2 =2G. I.e. the area B of the large brown square is twice the area G of the green square. p q G B

4. 2015-5-12 Calculus and Elementary Analysis Integers, Rational and Real Numbers Theorem There are not rational numbers r such that r 2 =2. Proof (cont’d) p q p-q A A I Furthermore, p and q are smallest integers such that the area B of the large brown square is twice the area G of the smaller green square, B=2G. p q Now move a copy of the green square to the upper right hand corner of the larger square. The two squares marked by A in the picture have the same area A. Furthermore, by the assumptions, I=2A. This is impossible, since B and G were the smallest squares with integer side lengths such that B=2G. The intersection I of the two copies of the green square is the square I with the area I.

5. 2015-5-12 Calculus and Elementary Analysis 1.2.2 Upper and Lower Bounds Let A be a non-empty set of real numbers. Definition 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.

6. 2015-5-12 Calculus and Elementary Analysis 1.2.3 Bounds of Integers, Rationals and Reals Observations Any non-empty bounded set of integers has the largest (and the smallest) element which is also an integer. 1 2 Consider the set of rational numbers r such that r 2 ≤ 2. This set is clearly bounded both from the above and from below. Observe that the set of rational upper bounds for the elements of this set does not have a smallest element. This follows from the previous considerations showing that is not rational. This means that the set of rational numbers is not complete in the sense that the set of rational upper bounds of a bounded set does not necessarily have the smallest rational upper bound.

7. 2015-5-12 Calculus and Elementary Analysis 1.2.4 Supremum Definition 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. 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 Completeness of Real Numbers

8. 2015-5-12 Calculus and Elementary Analysis Infimum Definition 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. 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

9. 2015-5-12 Calculus and Elementary Analysis 1.2.5 Characterization of the Supremum (1) Proof Theorem

10. 2015-5-12 Calculus and Elementary Analysis Characterization of the Supremum (2) Theorem Proof Cont’d For a non-empty set A:

11. 2015-5-12 Calculus and Elementary Analysis 1.2.6 Characterization of the Infimum Theorem The proof of this result is a repetition of the argument the previous proof for the supremum. For a non-empty set A:

12. 2015-5-12 Calculus and Elementary Analysis 1.2.7 Usage of the Characterizations Example Claim Proof of the Claim 1 2 12 and