NOC:Real Analysis - I

Lecture 1 - WEEK 1 - INTRODUCTION

Lecture 2 - Why study Real Analysis

Lecture 3 - Square root of 2

Lecture 4 - Wason's selection task

Lecture 5 - Zeno's Paradox

Lecture 6 - Basic set theory

Lecture 7 - Basic logic

Lecture 8 - Quantifiers

Lecture 9 - Proofs

Lecture 10 - Functions and relations

Lecture 11 - Axioms of Set Theory

Lecture 12 - Equivalence relations

Lecture 13 - What are the rationals

Lecture 14 - Cardinality

Lecture 15 - WEEK 2 - INTRODUCTION

Lecture 16 - Field axioms

Lecture 17 - Order axioms

Lecture 18 - Absolute value

Lecture 19 - The completeness axiom

Lecture 20 - Nested intervals property

Lecture 21 - NIP+AP⇒ Completeness

Lecture 22 - Existence of square roots

Lecture 23 - Uncountability of the real numbers

Lecture 24 - Density of rationals and irrationals

Lecture 25 - WEEK 3 - INTRODUCTION

Lecture 26 - Motivation for infinite sums

Lecture 27 - Definition of sequence and examples

Lecture 28 - Definition of convergence

Lecture 29 - Uniqueness of limits

Lecture 30 - Achilles and the tortoise

Lecture 31 - Deep dive into the definition of convergence

Lecture 32 - A descriptive language for convergence

Lecture 33 - Limit laws

Lecture 34 - Subsequences

Lecture 35 - Examples of convergent and divergent sequences

Lecture 36 - Some special sequences-CORRECT

Lecture 37 - Monotone sequences

Lecture 38 - Bolzano-Weierstrass theorem

Lecture 39 - The Cauchy Criterion

Lecture 40 - MCT implies completeness

Lecture 41 - Definition and examples of infinite series

Lecture 42 - Cauchy tests-Corrected

Lecture 43 - Tests for convergence

Lecture 44 - Erdos_s proof on divergence of reciprocals of primes

Lecture 45 - Resolving Zeno_s paradox

Lecture 46 - Absolute and conditional convergence

Lecture 47 - Absolute convergence continued

Lecture 48 - The number e

Lecture 49 - Grouping terms of an infinite series

Lecture 50 - The Cauchy product

Lecture 51 - WEEK 5 - INTRODUCTION

Lecture 52 - The role of topology in real analysis

Lecture 53 - Open and closed sets

Lecture 54 - Basic properties of adherent and limit points

Lecture 55 - Basic properties of open and closed sets

Lecture 56 - Definition of continuity

Lecture 57 - Deep dive into epsilon-delta

Lecture 58 - Negating continuity

Lecture 59 - The functions x and x2

Lecture 60 - Limit laws

Lecture 61 - Limit of sin x_x

Lecture 62 - Relationship between limits and continuity

Lecture 63 - Global continuity and open sets

Lecture 64 - Continuity of square root

Lecture 65 - Operations on continuous functions

Lecture 66 - Language for limits

Lecture 67 - Infinite limits

Lecture 68 - One sided limits

Lecture 69 - Limits of polynomials

Lecture 70 - Compactness

Lecture 71 - The Heine-Borel theorem

Lecture 72 - Open covers and compactness

Lecture 73 - Equivalent notions of compactness

Lecture 74 - The extreme value theorem

Lecture 75 - Uniform continuity

Lecture 76 - Connectedness

Lecture 77 - Intermediate Value Theorem

Lecture 78 - Darboux continuity and monotone functions

Lecture 79 - Perfect sets and the Cantor set

Lecture 80 - The structure of open sets

Lecture 81 - The Baire Category theorem

Lecture 82 - Discontinuities

Lecture 83 - Classification of discontinuities and monotone functions

Lecture 84 - Structure of set of discontinuities

Lecture 85 - WEEK 8 and 9 - INTRODUCTION

Lecture 86 - Definition and interpretation of the derivative

Lecture 87 - Basic properties of the derivative

Lecture 88 - Examples of differentiation

Lecture 89 - Darboux_s theorem

Lecture 90 - The mean value theorem

Lecture 91 - Applications of the mean value theorem

Lecture 92 - Taylor's theorem NEW

Lecture 93 - The ratio mean value theorem and L_Hospital_s rule

Lecture 94 - Axiomatic characterisation of area and the Riemann integral

Lecture 95 - Proof of axiomatic characterization

Lecture 96 - The definition of the Riemann integral

Lecture 97 - Criteria for Riemann integrability

Lecture 98 - Linearity of integral

Lecture 99 - Sets of measure zero

Lecture 100 - The Riemann-Lebesgue theorem

Lecture 101 - Consequences of the Riemann-Lebesgue theorem

Lecture 102 - WEEK 10 and 11 - INTRODUCTION

Lecture 103 - The fundamental theorem of calculus

Lecture 104 - Taylor's theorem-Integral form of remainder

Lecture 105 - Notation for Taylor polynomials

Lecture 106 - Smooth functions and Taylor series

Lecture 107 - Power series

Lecture 108 - Definition of uniform convergence

Lecture 109 - The exponential function

Lecture 110 - The inverse function theorem

Lecture 111 - The Logarithm

Lecture 112 - Trigonometric functions

Lecture 113 - The number Pi

Lecture 114 - The graphs of sin and cos

Lecture 115 - The Basel problem

Lecture 116 - Improper integrals

Lecture 117 - The Integral test

Lecture 118 - Weierstrass approximation theorem

Lecture 119 - Bernstein Polynomials

Lecture 120 - Properties of Bernstein polynomials

Lecture 121 - Proof of Weierstrass approximation theorem