Fundamentals of Real Analysis: Core Concepts and Techniques

REAL ANALYSIS CORE-3 UNIT-1 SRI SMRUTI RANJAN SAHOO Assistant Professor of Mathematics Department of Mathematics B.J.B AUTONOMOUS COLLEGE, BHUBANESWAR. SRI SMRUTI RANJAN SAHOO ( Assistant REAL ANALYSISCORE-3UNIT-1 1 / 16

Introduction The Calculusis one of mankinds most significant scientific achievements, transforming previously intractable physical problems into often routine calculations. It was developed in the late 17th century by Newton, when developing his laws of motion and gravitation, and Leibniz, who developed the notation we still use today. Analysis is the branch of mathematics that underpins the theory behind the calculus, placing it on a firm logical foundation through the introduction of the notion of a limit. Real analysis is the area of mathematics dealing with real numbers and the analytic properties of real-valued functions and sequences. Real Analysis is what mathematicians would call rigorous version of Calculus SRI SMRUTI RANJAN SAHOO ( Assistant REAL ANALYSISCORE-3UNIT-1 2 / 16

Techniques of Proof Direct Proof Proof by Induction Proof by Contradiction Proof by Contrapositive SRI SMRUTI RANJAN SAHOO ( Assistant REAL ANALYSISCORE-3UNIT-1 3 / 16

Law of Trichotomy The Law of Trichotomy states that every real number is either positive, negative, or zero. For arbitrary real numbers x and y, exactly one of x < y, y < x, or x = y applies. SRI SMRUTI RANJAN SAHOO ( Assistant REAL ANALYSISCORE-3UNIT-1 4 / 16

Upper Bound of a set S Definition Let S R be non-empty. We say that a real no is an upper bound of a set S if, for each x S, we have x . In terms of quantifiers, is an upper bound of Sif x S(x ) Geometrically, this means that elements of Sare to the left of . SRI SMRUTI RANJAN SAHOO ( Assistant REAL ANALYSISCORE-3UNIT-1 5 / 16

Is Upper Bound Unique? This Geometry shows that we can have more than one upper bound of a set S. So upper bound of a set need not to be unique. Figure: Upper Bound of S SRI SMRUTI RANJAN SAHOO ( Assistant REAL ANALYSISCORE-3UNIT-1 6 / 16

Least Upper Bound of S A Set Sis said to be bounded above if an upper bound of S. Definition Let = / S R be bounded above. A real number R is said to be a least upper bound of Sif, (i) is an upper bound of Sand (ii) if is an upper bound of S, then Equivalent Definition A real number R is the least upper bound of S iff (i) is an upper bound of Sand (ii)if < , then is not an upper bound of Si.e. if < , then there exists c S such that c > SRI SMRUTI RANJAN SAHOO ( Assistant REAL ANALYSISCORE-3UNIT-1 7 / 16

Geometrical Interpretation of LUB Figure: Least Upper Bound of S SRI SMRUTI RANJAN SAHOO ( Assistant REAL ANALYSISCORE-3UNIT-1 8 / 16

- Definition of Least Upper Bound A real number R is the least upper bound of S if and only if (i) is an upper bound of S (ii) for any > 0, is not an upper bound of S, i.e., there exist a s S such that s > . Geometrically you can compare with the previous definition. Example Let S = (0,1). Then Lub of S is 1. SRI SMRUTI RANJAN SAHOO ( Assistant P rofessor of Mathematics Department of MatB.J.B AUTONOMOUShematics REAL ANALYSISCORE-3UNIT-1 9 / 16

LUB Property(Completeness Property) LUB Property of R Given any non-empty subset Sof R which is bounded above in R, there exist R such that = lub S. SRI SMRUTI RANJAN SAHOO ( Assistant REAL ANALYSISCORE-3UNIT-1 10 / 16

Applications of LUB Property The Archimedean Property Definition 1 : Given any x R, there exist n N such that n > x. Definition 2 : Given any x, y R with x > 0, there exits n N such that nx > y The first definition gives N is unbounded above. The second definition tells about measurement in some units. SRI SMRUTI RANJAN SAHOO ( Assistant REAL ANALYSISCORE-3UNIT-1 11 / 16

Greatest Integer Function Let x R. Then there exists a unique m Z such that m x < m+ 1. Proof : Construct S = {k Z : k x}. Now S /= , otherwise k Z(k > x). Then choosing k = n, n N we get n > x implies n < x, n N implies N is bounded above a contradiction. Then by LUB Property LUB of S, let LUB(S) = . Then 1 is not an upper bound, hence there is a k Ssuch that 1 < k. Then k x. Next to show x < k + 1. If not then k + 1 x implies k + 1 S. Hence k + 1 < which is a contradiction. Try to think if there exist another msuch that m x< m + 1, whether this mand k areequal or not. If not then by law of trichotomy and by the inequality we have obtained we will reach to a contradiction. SRI SMRUTI RANJAN SAHOO ( Assistant REAL ANALYSISCORE-3UNIT-1 12 / 16

Consequence of LUB Property. Density of Q is obtained. Z is unbounded Weierstrass Completeness Principle Cantor s Completeness Principle Bolzano-Weierstrass Theorem Cauchy Completeness Principle Intermediate Value Theorem Heine- Borel Theorem. SRI SMRUTI RANJAN SAHOO ( Assistant REAL ANALYSISCORE-3UNIT-1 13 / 16

Summary Basically Here I have gave you an Idea how we can use this LUB Property of R to visualize the things that we usually use in mathematics. LUB Propery or Completeness property is the base for all the theorems and proofs of the subject SRI SMRUTI RANJAN SAHOO ( Assistant REAL ANALYSISCORE-3UNIT-1 14 / 16

For Further Reading I A Basic Course in Real Analysis. By S. Kumaresan, Ajit Kumar Intoduction to Real Analysis. By Robert G, Bartle, Donald R. Sherbert ElementaryAnalysis : The Theoryof CalculusBy Kenneth A. Ross Principles of Mathematical Analysis. By Walter Rudin SRI SMRUTI RANJAN SAHOO ( Assistant REAL ANALYSISCORE-3UNIT-1 15 / 16

PURE MATHEMATICS IS, IN ITS WAY, THE POETRY OF LOGICAL IDEAS (ALBERT EINSTEIN) THANK YOU SRI SMRUTI RANJAN SAHOO ( Assistant REAL ANALYSISCORE-3UNIT-1 16 / 16