Introduction to Real Analysis
Introduction to Real Analysis. Instructor: Prof. William Kinney, Math and Computer Science, Bethel University. This is a collection of course lectures, dealing with topics on Real Analysis: basic set theory, the real numbers, basic topology, sequences and series of real numbers, limits and continuity, differentiation, sequences and series of functions, and Riemann integration.
Lecture 35 - Sup Norm and Metric on C[a,b], Sequence Space, Open and Closed Sets
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Lecture 01 - Overview, Mean Value Theorem, Sqrt(2) is Irrational
Lecture 02 - Completeness Axiom, Counterexamples, Ordered Fields
Lecture 03 - Irrational Numbers, Supremums, Completeness, Sqrt(2) Exists
Lecture 04 - Cardinality, Cantor, Continuum Hypothesis, Ping Pong Ball Conundrum
Lecture 05 - Archimedean Property of R, Cantor's Theorem, Sequences, Crazy Functions
Lecture 06 - Bounded Sequences, Monotone Sequences, Limits of Sequences
Lecture 07 - Monotone Convergence, Bolzano-Weierstrass, Cauchy Sequences
Lecture 08 - Subsequences, Bolzano-Weierstrass, Cauchy Criterion, Limsup and Liminf
Lecture 09 - Recursive Sequences, Limit Superior and Inferior Definitions and Properties
Lecture 10 - Epsilon Delta Definition of Limit of Function, Proofs, Deleted Neighborhoods
Lecture 11 - Continuity and Intermediate Value Theorem (Climbing Monk Story)
Lecture 12 - Limits Involving Infinity, Continuity, Intermediate and Extreme Values
Lecture 13 - Prove Extreme Value Theorem, Intermediate Value Property, Uniform Continuity
Lecture 14 - Uniform Continuity Non-Examples, Variation of a Function
Lecture 15 - Uniform Continuity, Monotone Functions, Devil's Staircase, Derivatives
Lecture 16 - Mean Value Theorem: Statement, Basic Examples, and Proof
Lecture 17 - Mean Value Theorem Corollaries, Definition of Riemann Integral
Lecture 18 - Optimization Exs, Step Functions are Riemann Integrable
Lecture 19 - Conditions for Riemann Integrability
Lecture 20 - Prove Cubic Has Unique Real Root using MVT, Riemann Integrability
Lecture 21 - Convergence of Riemann Sums, Fundamental Theorem of Calculus
Lecture 22 - Review Properties of Integrals and Fundamental Theorem of Calculus
Lecture 23 - Fund Thm Calculus, Integration by Parts, Average Value, Telescoping Series
Lecture 24 - Convergence of Series, Geometric Series, p-Series Test, Divergence Test
Lecture 25 - Comparison and Limit Comparison Test, Absolute Convergence, Ratio and Root Test
Lecture 26 - Integrals of Step Functions, Various Series Facts, Sequences of Functions
Lecture 27 - Pointwise and Uniform Convergence of Sequences and Series of Functions Examples
Lecture 28 - Taylor Series for Sin(x), Cos(x), e^(x), Uniform Convergence
Lecture 29 - The Most Beautiful Equation in the World, Taylor Series Calculations
Lecture 30 - Review for Exam 3: Series Tricks, Uniform Convergence, Weierstrass M-Test
Lecture 31 - Open Sets on the Real Line, Continuity and Preimages of Open Intervals
Lecture 32 - Open and Closed Sets in the Real Line and in the Plane
Lecture 33 - Euclidean Metric, Triangle Inequality, Metric Spaces, Compact Sets
Lecture 34 - P-Norm, Sup Norm, Continuity and Preimages, Images of Compact Sets
Lecture 35 - Sup Norm and Metric on C[a,b], Sequence Space, Open and Closed Sets
Lecture 36 - Chaos, Logistic Map, Period Doubling, Symbol Space, Shift Map
Lecture 37 - Taylor Series, Cantor Sets and Logistic Map, Review Topology