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 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 |