Mathematical Analysis

Mathematical Analysis
Real Analysis lecture photos:



Sofia Kovalevskaya 1850 – 1891


Cathleen Morawetz – 1923 –

I find that I may have emphasized the need to escape from the devils of mathematics to embark on the pleasures of the real world. But it works both ways, and sometimes the devils of the real world drive one into the pleasures of studying mathematics.

Mathematicians on Creativity by Peter Borwein

Mathematical Analysis


Below is a set of notes on Mathematical Analysis

Chapter 1 : Logic and Proof

Section A:

Propositional Logic


userfiles/Section A Propositional Logic(1).pdf userfiles/Exercise 1(a)(1).pdf
Section B:

Algebra of Propositions


userfiles/Section B Algebra of Propositions.pdf userfiles/Exercise 1b.pdf userfiles/Complete solutions to Exercise 1b.pdf
 Section C:



 userfiles/Section C Implication.pdf  userfiles/Exercise 1(c).pdf  userfiles/Complete solutions to Exercise 1(c)(1).pdf
 Section D:

Introduction to Proof


 userfiles/Section D Introduction to Proof.pdf  userfiles/Exercise 1d(1).pdf  userfiles/Complete solutions to Exercise 1d(1).pdf
 Section E:



 userfiles/SECTION E PROOFs.pdf  userfiles/Exercise 1.pdf
 Section F:

Proofs by Contradiction


 userfiles/SECTION F PROOF BY CONTRADICTION.pdf  userfiles/Exercise 1F(1).pdf  userfiles/Complete Solutions to Exercise 1f(1).pdf
 Section G:

Principle of Mathematical Induction


 userfiles/Section G Principle Of Mathematical Induction.pdf  userfiles/Exercise 1g.pdf  userfiles/Complete Solutions to Exercise 1g.pdf
 Section H:

Further Mathematical Induction


 userfiles/SECTION H Further Mathematical Induction.pdf  userfiles/Exercise 1h.pdf  userfiles/Complete Solutions to Exercise 1h.pdf

Notes on inequalities

SECTION D Inequalities     Exercise 4d

Video on Inequalities Part I

Inequalities Part II

Notes on Inequalities


SECTION E Modulus Function

Video on Modulus Function

Notes on modulus function

Video on Inequality Properties of the modulus function 

Notes on Inequality Properties of the modulus function

Exercise 4e

Chapter 2 : Infinite Series

Section A:

Introduction to Infinite Series

Video on Introduction to Infinite Series

Video on Examples of Infinite Series

Video on examples of Geometric Series



userfiles/Introduction to Infinite Series(1).pdf Exercise 2a  Solutions 2a
Section B:

Telescoping and Harmonic Series

Video on finding the sum by partial  fractions

Video on Harmonic Series


userfiles/Telescoping and Harmonic Series.pdf userfiles/Exercise 2b(1).pdf  Solutions 2b
 Section C:

Properties of Convergent and Divergent Series


Section 2C Convergent and Divergent Properties

Section 2C

Rules of convergence

Rules of convergence 2

 userfiles/Exercise 2c(1).pdf Solutions 2c
 Section D:

Comparison Test and p-series


userfiles/Section 2D Comparison Test and p-series.pdf  userfiles/Exercise 2d.pdf Solutions 2d
 Section E:

Ratio Test


 userfiles/Section 2E Ratio Test.pdf  userfiles/Exercise 2e.pdf Solutions 2e
 Section F:

Integral Test


 userfiles/Section 2F Integral Test.pdf  userfiles/Exercise 2f.pdf Solutions 2f
 Section G:

Alternating Series


 userfiles/Section 2G Alternating Series.pdf  userfiles/Exercise 2g.pdf  Solutions 2g
 Section H to be added later.

Section I:

Power Series


 userfiles/Section 2I Power Series.pdf  userfiles/Exercise 2i.pdf

Notes on Pointwise and Uniform Convergence


Sequences of Functions

Pointwise Convergence

 Section B Uniform Convergence
Section C

Section D

Taylor Series and Largest Error

Chapter 3 : Functions

Section A

Introduction to Functions

Videos on

Defintion of a function

Notes on Defn of a function

Range of a function part I

Range of a function part II

Notes on Range of a function


Chapter 3A Functions userfiles/Exercise 3a(1).pdf
Section B:

Injective and Surjective

Videos on

Injective Functions

Notes on Injective functions

Surjective functions

Notes on Surjective functions

SECTION 3B Injective and Surjective functions userfiles/Exercise 3b(1).pdf
Section C:

Inverse Functions


Bijective Functions

Inverse Functions



Section 3c Inverse functions userfiles/Exercise 3c(1).pdf
Section D:

Composite Functions


userfiles/SECTION D Composite Functions.pdf userfiles/Exercise 3d(1).pdf
Section E:

Properties of Composite Functions


userfiles/SECTION E Properties of Composite Functions.pdf userfiles/Exercise 3e(1).pdf
Section F:

Graphs of Functions


userfiles/SECTION F Graphs of Functions.doc userfiles/Exercise 3f.pdf

Chapter 5 : Sequences and Real Numbers

Section A:

Upper and Lower Bounds


userfiles/Section A Upper and Lower Bounds.pdf userfiles/Exercise 5a(1).pdf
Section B:

Properties of Upper and Lower Bounds

userfiles/5B Properties of Upper and Lower Bounds.pdf userfiles/Exercise 5b(1).pdf
Section C:

Limit of a Sequence

Video on

Introduction to limit of a function

Proving limit of a function


userfiles/Section C The Limit of a Sequence.pdf Exercise 5c userfiles/Complete Solutions to Exercise 5c(1).pdf
Section D:

Properties of the Limit of a Sequence


Section D Properties of The Limit of a Sequence userfiles/Exercise 5d.pdf userfiles/Complete Solutions to Exercise 5d.pdf
Section E:

Evaluating Limits


userfiles/Section E Evaluating Limits.pdf userfiles/Exercise 5e.pdf userfiles/Complete Solutions to Exercise 5e.pdf
Section F:

Monotone Sequences


userfiles/Section F Monotone Sequences.pdf userfiles/Exercise 5f.pdf userfiles/Complete solutions to Exercise 5f.pdf
Section G:



userfiles/Section G Subsequences.pdf

Exercise 7H

Complete Solutions to 7h

Exercise 7i

Complete solutions to Exercise 7i.pdf