Lecture 39 of 123Week 4
10.3 The Cauchy Criterion
Prof. Jaikrishnan J · IIT Palakkad
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- Real Analysis I, IIT Palakkad (at NPTEL)
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All the lectures
- 1.1 WEEK 1 INTRODUCTION
- 1.2 Why study Real Analysis
- 1.3 Square root of 2
- 1.4 Wason's selection task
- 1.5 Zeno's Paradox
- 2.1 Basic set theory
- 2.2 Basic logic
- 2.3 Quantifiers
- 2.4 Proofs
- 2.5 Functions and relations
- 3.1 Axioms of Set Theory
- 3.2 Equivalence relations
- 3.3 What are the rationals
- 3.4 Cardinality
- WEEK 2 INTRODUCTION
- 4.1 Field axioms
- 4.2 Order axioms
- 4.3 Absolute value
- 5.1 The completeness axiom
- 5.2 Nested intervals property
- 6.1 NIP+AP⇒ Completeness
- 6.2 Existence of square roots
- 6.3 Uncountability of the real numbers
- 6.4 Density of rationals and irrationals
- WEEK 3 INTRODUCTION
- 7.1 Motivation for infinite sums
- 7.2 Definition of sequence and examples
- 7.3 Definition of convergence
- 7.4 Uniqueness of limits
- 7.5 Achilles and the tortoise
- 8.1 Deep dive into the definition of convergence
- 8.2 A descriptive language for convergence
- 8.3 Limit laws
- 9.1 Subsequences
- 9.2 Examples of convergent and divergent sequences
- 9.3 Some special sequences-CORRECT
- 10.1 Monotone sequences
- 10.2 Bolzano-Weierstrass theorem
- 10.3 The Cauchy Criterion
- 10.4 MCT implies completeness
- 11.1 Definition and examples of infinite series
- 11.2 Cauchy tests-Corrected
- 11.3 Tests for convergence
- 11.4 Erdos_s proof on divergence of reciprocals of primes
- 11.5 Resolving Zeno_s paradox
- 12.1 Absolute and conditional convergence
- 12.2 Absolute convergence continued
- 12.3 The number e
- 12.4 Grouping terms of an infinite series
- 12.5 The Cauchy product
- WEEK 5 - INTRODUCTION
- 13.1 The role of topology in real analysis
- 13.2 Open and closed sets
- 13.3 Basic properties of adherent and limit points
- 13.4 Basic properties of open and closed sets
- 14.1 Definition of continuity
- 14.2 Deep dive into epsilon-delta
- 14.3 Negating continuity
- 15.1 The functions x and x2
- 15.2 Limit laws
- 15.3 Limit of sin x_x
- 15.4 Relationship between limits and continuity
- 15.5 Global continuity and open sets
- 15.6 Continuity of square root
- 15.7 Operations on continuous functions
- 16.1 Language for limits
- 16.2 Infinite limits
- 16.3 One sided limits
- 16.4 Limits of polynomials
- 17.1 Compactness
- 17.2 The Heine-Borel theorem
- 17.3 Open covers and compactness
- 17.4 Equivalent notions of compactness
- 18.1 The extreme value theorem
- 18.2 Uniform continuity
- 19.1 Connectedness
- 19.2 Intermediate Value Theorem
- 19.3 Darboux continuity and monotone functions
- 20.1 Perfect sets and the Cantor set
- 20.2 The structure of open sets
- 20.3 The Baire Category theorem
- 21.1 Discontinuities
- 21.2 Classification of discontinuities and monotone functions
- 21.3 Structure of set of discontinuities
- WEEK 8 & 9 - INTRODUCTION
- 22.1 Definition and interpretation of the derivative
- 22.2 Basic properties of the derivative
- 22.3 Examples of differentiation
- 23.1 Darboux_s theorem
- 23.2 The mean value theorem
- 23.3 Applications of the mean value theorem
- 24.1 Taylor's theorem NEW
- 24.2 The ratio mean value theorem and L_Hospital_s rule
- 25.1 Axiomatic characterisation of area and the Riemann integral
- 25.2 Proof of axiomatic characterization
- 26.1 The definition of the Riemann integral
- 26.2 Criteria for Riemann integrability
- 26.3 Linearity of integral
- 27.1 Sets of measure zero
- 27.2 The Riemann-Lebesgue theorem
- 27.3 Consequences of the Riemann-Lebesgue theorem
- WEEK 10 & 11 - INTRODUCTION
- 28.1 The fundamental theorem of calculus
- 28.2 Taylor's theorem-Integral form of remainder
- 28.3 Notation for Taylor polynomials
- 28.4 Smooth functions and Taylor series
- 29.1 Power series
- 29.2 Definition of uniform convergence
- 31.1 The exponential function
- 31.2 The inverse function theorem
- 31.3 The Logarithm
- 32.1 Trigonometric functions
- 32.2 The number Pi
- 32.3 The graphs of sin and cos
- 33.1 The Basel problem
- 34.1 Improper integrals
- 34.2 The Integral test
- 35.1 Weierstrass approximation theorem
- 35.2 Bernstein Polynomials
- 35.3 Properties of Bernstein polynomials
- 35.4 Proof of Weierstrass approximation theorem
- Live Session 04-10-2020
- Live Session 10-11-2020