This course in Real Analysis aims to teach students the concepts of supremum, completeness of real numbers, equinumerosity, countability of rational numbers, uncountability of real numbers, sequences, convergence, limits, series, open subsets, limit points, compact sets, Heine-Borel Theorem, connected sets, continuity, differentiability, derivatives, Mean Value Theorem, L'Hospital's Rule, Riemann Integrability, and the Fundamental Theorem of Calculus. The course uses a combination of lectures, examples, and homework assignments to help learners develop skills in analyzing and understanding real numbers and functions. This course is intended for students or individuals interested in advanced mathematical analysis and calculus.
Overview
Syllabus
Real Analysis | The Supremum and Completeness of ℝ.
Real Analysis | The density of Q and other consequences of the Axiom of Completeness..
Real Analysis | Equinumerosity.
Real Analysis | The countability of the rational numbers..
Real Analysis | The uncountability of ℝ.
Real Analysis | ℝ and P(ℕ).
Real Analysis | Sequences and the ε-N definition of convergence..
Real Analysis| Three limits of sequences by the definition..
Real Analysis | A convergent sequence is bounded..
Real Analysis | Algebraic Properties of Limits.
Real Analysis | The monotone sequence theorem..
Real Analysis | Monotone sequence theorem example..
Real Analysis | Monotone sequence theorem example 2..
Real Analysis | A first look at series..
Real Analysis | The Cauchy Condensation Test.
A nice limit with a trick..
Real Analysis | Subsequences.
Real Analysis | Cauchy Sequences.
Real Analysis | Cauchy Criterion for Series.
Real Analysis | Proving some series tests..
Real Analysis | Rearrangements of absolutely convergent series..
Real Analysis | Open subsets of ℝ..
Real Analysis | The limit point of a set A⊆ℝ.
Real Analysis | Isolated points.
Real Analysis | Closed Sets.
Real Analysis | The closure of a set..
Real Analysis | Compact set of real numbers..
Real Analysis | Nested compact sets..
Real Analysis | The Heine-Borel Theorem.
Real Analysis | If [a,b] is compact so is any closed and bounded set..
Real Analysis | Perfect Sets.
Real Analysis | Connected Sets.
Real Analysis | Precise definition of a limit..
Real Analysis | Sequential limits in functions..
Real Analysis | Showing a function is (dis)continuous..
Real Analysis | The continuous image of a compact set..
Real Analysis | Intro to uniform continuity..
Real Analysis | Showing a function is not uniformly continuous..
Real Analysis | Uniform continuity and compact sets..
Real Analysis | The uniform continuity of sqrt(x)..
Real Analysis | Topological continuity.
Real Analysis | Continuity, connected sets, and the IVT..
Real Analysis | Introduction to differentiability..
Real Analysis | Derivative Rules.
Real Analysis | Where are extreme values?.
Real Analysis | The Mean Value Theorem.
Real Analysis | The Generalized Mean Value Theorem and One part of L'Hospital's rule..
Real Analysis | L'Hospital's Rule (∞/∞ - case).
Real Analysis | Pointwise convergence of sequences of functions..
Real Analysis | Motivating uniform convergence.
Real Analysis | Uniform Convergence and Continuity.
Real Analysis | Uniform Convergence and Differentiability.
Real Analysis | Series of Functions.
Real Analysis | Partitions and upper/lower sums..
Real Analysis | Refinements of partitions..
Real Analysis | Riemann Integrability.
Real Analysis | An important property of integration..
Real Analysis | Sequences of functions and integration..
Real Analysis | The Fundamental Theorem of Calculus.
Real Analysis homework on the Putnam?.
Taught by
Michael Penn
