MA 101 (Mathematics-I)• S. R. Ghorpade and B. V. Limaye,A Course in Calculus and Real Analysis,...

Transcript of MA 101 (Mathematics-I)• S. R. Ghorpade and B. V. Limaye,A Course in Calculus and Real Analysis,...

IntroductionSuprema and Infima

Archimedean property

MA 101 (Mathematics-I)

Anupam Saikia and Rupam BarmanDepartment of Mathematics

IIT Guwahati

A. Saikia & R. Barman MA-101 (2019)


Outline of the Course

Two Topics:

• Single variable calculusWill be taught as the first part of the course. TotalNumber of Lectures= 21 and Tutorials = 7.

• R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis, WileyIndia, 4th Edition, 2014.

• G. B. Thomas, Jr. and R. L. Finney, Calculus and AnalyticGeometry, 6th/ 9th Edition, Narosa/ Pearson Education India, 1996.

• S. R. Ghorpade and B. V. Limaye, A Course in Calculus and RealAnalysis, 5th Indian Reprint, Springer, 2010.

• Multivariable CalculusWill be taught as the second part of the course.



Outline of the CourseInstructors (Single variable calculus):Prof. Anupam Saikia and Dr. Rupam Barman

Course webpage (Single variable calculus):http://www.iitg.ernet.in/rupam/

• For Lecture Divisions and Tutorial Groups, LectureVenues, Tutorial Venues and Class & Exam Time Tables,See Intranet Academic Section Website.

• Tutorial problem sheets will be uploaded in the coursewebpage (also available at the xerox shops: Lecture Halland Core-1). You are expected to try all the problems inthe problem sheet before coming to the tutorial class.Do not expect the tutor to solve completely all theproblems given in the tutorial sheet.




Attendance Policy

Attendance in all lecture and tutorial classes is compulsory.

As per Institute guidelines, students who do not meet 75%attendance requirement in the course will NOT be allowed towrite the end semester examination and will be awarded F(Fail) grade in the course.

In this course we will strictly follow the Institute guidelines onattendance policy. There will be 42 classes of this course.Therefore, students must attend at least 30 classes.

(Refer: B.Tech. Ordinance Clause 4.1)




Marks distribution:

1. Quiz: 20 percentage (Two quizes: Quiz-1 from singlevariable calculus and Quiz-2 from multivariable calculus)

2. QUIZ-1 will be held on August 23rd, 2019

3. Mid-term: 30 percentage

4. End-term: 50 percentage (20% will be on single variablecalculus)

No make up test for Quizzes and Mid Semester Examination.

Do preserve your (evaluated) answer scripts of Quizzes andMid Semester Examination of MA101 till the completion ofthe Course Grading.



Introduction

We denote the set of natural numbers by N, the set ofintegers by Z, and the set of rational numbers by Q, and weassume familiarity with each of these sets:

• N = {1, 2, 3, . . .}• Z = {0,±1,±2,±3, . . .}• Q = {m/n : m ∈ Z, n ∈ N}

The Well-Ordering Property of N states that everynonempty subset of N has a least element.

That is, given a nonempty subset S of N, there exists m ∈ Ssuch that m ≤ k for all k ∈ S . The element m is the leastelement of S .



Introduction


• N = {1, 2, 3, . . .}• Z = {0,±1,±2,±3, . . .}• Q = {m/n : m ∈ Z, n ∈ N}





Introduction


• N = {1, 2, 3, . . .}• Z = {0,±1,±2,±3, . . .}• Q = {m/n : m ∈ Z, n ∈ N}





The set of real numbers, denoted by R, is best described moregeometrically by setting up a one-to-one correspondence withpoints of a line that stretches infinitely in both directions. Amore formal and technically sound definition requires morebackground; we postpone it for later. Instead, assuming thisgeometrical correspondence, we list three sets of axioms thatthe set of real numbers follow:



Field Axioms:

1 (Associative laws) x + (y + z) = (x + y) + z andx(yz) = (xy)z for all x , y , z ∈ R

2 (Commutative laws)x + y = y + x and xy = yx for all x , y ∈ R

3 (Identities)x + 0 = x = 0 + x and x · 1 = x = 1 · x for all x ∈ R

4 (Inverses) x + (−x) = 0 = (−x) + x for all x ∈ R andx · 1

x= 1 = 1

x· x for all x ∈ R \ {0}

5 (Distributive laws)x(y+z) = xy+xz and (x+y)z = xz+yz for all x , y , z ∈ R



Order Axioms:

1 For each x , y ∈ R, exactly one of x > y , x = y , x < yholds

2 If x ≥ y , then x + z ≥ y + z for all z ∈ R3 If x ≥ y and z ≥ 0, then xz ≥ yz

From the Order Axioms one can derive the usual inequalitiessatisfies by the set of real numbers. One of the mostimportant properties dealing with inequalities is the following:

Property

If x + ε ≥ y holds for all ε > 0, then x ≥ y also holds.



Order Axioms:

1 For each x , y ∈ R, exactly one of x > y , x = y , x < yholds

2 If x ≥ y , then x + z ≥ y + z for all z ∈ R3 If x ≥ y and z ≥ 0, then xz ≥ yz

From the Order Axioms one can derive the usual inequalitiessatisfies by the set of real numbers. One of the mostimportant properties dealing with inequalities is the following:

Property

If x + ε ≥ y holds for all ε > 0, then x ≥ y also holds.



Absolute Value:The absolute value |a| of a real number a isdefined as max{a,−a}. In particular, |a| = | − a| ≥ 0 for alla ∈ R, with equality if and only if a = 0.

Two of the most significant properties satisfied by the absolutevalue function | · | are: (i) |ab| = |a||b| for each pair a, b ∈ R,and (ii) (Triangle Inequality) |a + b| ≤ |a|+ |b| for each paira, b ∈ R.

Geometrically, |a − b| denotes the distance between the realnumbers a and b. In particular, |a| denotes the distance of thereal number a from the origin.

The inequality |x − a| < ε easily translates toa − ε < x < a + ε and the inequality |x − a| > ε translates tox > a + ε or x < a − ε.



Bounded sets

Let S be a nonempty subset of R.(a) A real number u is called an upper bound of S if a ≤ u

for each a ∈ S . The set S is said to be bounded above ifit has an upper bound.

(b) A real number v is called a lower bound of S if a ≥ v foreach a ∈ S . The set S is said to be bounded below if ithas a lower bound.

(c) A set is said to be bounded if it is both bounded aboveand bounded below. A set is said to be unbounded if it isnot bounded.

Observation:S is bounded precisely when there is a realnumber M for which |a| ≤ M holds for every a ∈ S .



Bounded sets








Bounded sets








Bounded sets








Least upper bound and greatest lower bound

If S is bounded above, then a number u is said to be asupremum (or a least upper bound) of S if it satisfies thefollowing conditions:

(a) u is an upper bound of S

(b) if v is any upper bound of S , then u ≤ v .

If S is bounded below, then a number w is said to be aninfimum (or a greatest lower bound) of S if it satisfies thefollowing conditions:

(a) w is a lower bound of S

(b) if t is any lower bound of S , then t ≤ w .



Least upper bound and greatest lower bound

If S is bounded above, then a number u is said to be asupremum (or a least upper bound) of S if it satisfies thefollowing conditions:

(a) u is an upper bound of S

(b) if v is any upper bound of S , then u ≤ v .

If S is bounded below, then a number w is said to be aninfimum (or a greatest lower bound) of S if it satisfies thefollowing conditions:

(a) w is a lower bound of S

(b) if t is any lower bound of S , then t ≤ w .



Facts: A nonempty set S can have atmost one supremum.Similarly, a nonempty set S can have atmost one infimum.

If the supremum or the infimum of a set S exists, we willdenote them by

sup(S) and inf(S).





sup(S) and inf(S).





sup(S) and inf(S).



Maximum and minimum

A set S has a maximum when there exists M ∈ S such thata ≤ M for all a ∈ S . Observe that every nonempty set S canhave at most one maximum, and that the maximum (if itexists) is also the supremum of S . On the other hand, if thesupremum of S exists and sup(S) ∈ S , then sup(S) = max(S).

Analogous observation to greatest lower bound and tominimum of a set.



Maximum and minimum

A set S has a maximum when there exists M ∈ S such thata ≤ M for all a ∈ S . Observe that every nonempty set S canhave at most one maximum, and that the maximum (if itexists) is also the supremum of S . On the other hand, if thesupremum of S exists and sup(S) ∈ S , then sup(S) = max(S).

Analogous observation to greatest lower bound and tominimum of a set.



Completeness Axiom

It is not possible to prove on the basis of the field and orderproperties of R that every nonempty subset of R that isbounded above has a supremum in R.

It is a deep and fundamental property of the real numbersystem that this is indeed the case.

The completeness property of R states that everynonempty set of real numbers that has an upper bound alsohas a supremum in R; and that every nonempty set of realnumbers that has a lower bound also has an infimum in R.



Completeness Axiom






Completeness Axiom






Lemma (Property of supremum)

Let A be a nonempty set of real numbers, and suppose sup(A)exists. Then for every ε > 0, there exists a ∈ A such thatsup(A)− ε < a ≤ sup(A).

Lemma (Property of infimum)

Let A be a nonempty set of real numbers, and suppose inf (A)exists. Then for every ε > 0, there exists a ∈ A such thatinf (A) ≤ a < inf (A) + ε.



Lemma (Property of supremum)

Let A be a nonempty set of real numbers, and suppose sup(A)exists. Then for every ε > 0, there exists a ∈ A such thatsup(A)− ε < a ≤ sup(A).

Lemma (Property of infimum)

Let A be a nonempty set of real numbers, and suppose inf (A)exists. Then for every ε > 0, there exists a ∈ A such thatinf (A) ≤ a < inf (A) + ε.



Archimedean property: If x and y are positive real numbers,then there exists a positive integer n for which nx > y .

Example

Let S = { 1n

: n ∈ N}. We note that 0 < 1n≤ 1 for all n ∈ N.

Thus, 1 is an upper bound and since 1 ∈ S we havesup(S) = max(S) = 1.

Clearly, 0 is a lower bound of S . We can use Archimedeanproperty to prove that 0 is the greatest lower bound.




Example

Let S = { 1n







Example

Let S = { 1n






Density of rational numbers in R

The set of rational numbers is “dense” in R in the sense thatgiven any two real numbers there is a rational number betweenthem (in fact, there are infinitely many such rational numbers).

Theorem (The Density Theorem)

If x and y are any real numbers with x < y , then there existsa rational number r ∈ Q such that x < r < y .



Density of rational numbers in R

The set of rational numbers is “dense” in R in the sense thatgiven any two real numbers there is a rational number betweenthem (in fact, there are infinitely many such rational numbers).

Theorem (The Density Theorem)

If x and y are any real numbers with x < y , then there existsa rational number r ∈ Q such that x < r < y .


IntroductionSuprema and InfimaArchimedean property

MA 101 (Mathematics-I)• S. R. Ghorpade and B. V. Limaye,A Course in Calculus and Real Analysis,...

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Transcript of MA 101 (Mathematics-I)• S. R. Ghorpade and B. V. Limaye,A Course in Calculus and Real Analysis,...