Overview

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Dive into a comprehensive 15-hour course on Real Analysis, exploring fundamental concepts from the Supremum and Completeness of ℝ to the Fundamental Theorem of Calculus. Master key topics including equinumerosity, countability, sequences, series, limits, continuity, differentiability, and integration. Examine important theorems such as the Monotone Sequence Theorem, Heine-Borel Theorem, and Mean Value Theorem. Develop a deep understanding of topological concepts, uniform convergence, and Riemann integrability. Engage with challenging problems and gain proficiency in proving mathematical statements, preparing you for advanced mathematical analysis and potential Putnam-level problem-solving.

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?.