Real Analysis MOC (MATH 412)

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For a lecture note TOC, see the following, sorted by most recent lecture to newest:

File	Type
Lecture 33 - Supplemental Material, Weirstrass Function	Lecture Notes
Lecture 32 - Generalized MVT	Lecture Notes
Lecture 31 - Mean Value Theorem	Lecture Notes
Lecture 30 - Review Ch 3,4	Lecture Notes
Lecture 29 - Chain Rule	Lecture Notes
Lecture 28 - The Derivative	Lecture Notes
Lecture 27 - IVT	Lecture Notes
Lecture 26 - Uniform Continuity	Lecture Notes
Lecture 25 - Continuity	Lecture Notes
Lecture 24 - Intro to Functions	Lecture Notes
Lecture 23 - Perfect Sets and Connected Sets	Lecture Notes
Lecture 22 - Intro to Baire's Theorem	Lecture Notes
Lecture 21 - More Definitions on Topology	Lecture Notes
Lecture 20 - Finishing Cantor Set, Review of Ch. 2	Lecture Notes
Lecture 19 - Cantor Set	Lecture Notes
Lecture 18 - More on Double Sums	Lecture Notes
Lecture 17 - Double Sums	Lecture Notes
Lecture 15 - Cauchy Convergence	Lecture Notes
Lecture 16 - The Series Tests	Lecture Notes
Lecture 14 - Weierstrass Convergence Theorem	Lecture Notes
Lecture 13 - Series	Lecture Notes
Lecture 12 - Monotonic Sequences	Lecture Notes
Lecture 11 - Bounded Sequences, Limit Laws	Lecture Notes
Lecture 10 - More on Sequences	Lecture Notes
Lecture 9 - Introduction to Sequences and Series	Lecture Notes
Lecture 8 - Review	Lecture Notes
Cantor's Diagonalization Argument	Lecture Theorems
Lecture 7 - Cantor's Diagonalization Argument, More Topics	Lecture Notes
Lecture 6 - Cardinality	Lecture Notes
Lecture 5 - Some Consequences of Completeness	Lecture Notes
Lecture 4 - Continuing Our Construction of Reals	Lecture Notes
Lecture 3 - A Rigorous Construction of the Reals	Lecture Notes
Lecture 2 - Starting to Construct the Reals	Lecture Notes
Lecture 1 - Overview of the Course	Lecture Notes

Follow this guide to see what topics are covered in what sections in the actual textbook. Or you can just peruse the topics:

File	Section, Subsection
Absolute Convergence Test	Theorems
Addition on R (via Dedekind Cuts)	8 Additional Topics
Algebraic Continuity Theorem	Theorems
Algebraic Differentiability Theorem	Theorems
Algebraic Limit Theorem for Functional Limits	Theorems
Algebraic Limit Theorem for Series	Theorems
Alternating Series Test (AST)	Theorems
Balzano-Weierstrass Theorem	Theorems
Bounded For Sequences	Definitions
Bounds	Definitions
Cantor Set	3 Topology of R
Cantor's Approach	8 Additional Topics
Cantor's Diagonalization Method	Theorems
Cantor's Theorem	Theorems
Cardinality	Definitions
Cauchy Condensation Test	Theorems
Cauchy Criterion	Theorems
Cauchy Criterion for Series	Theorems
Cauchy Sequence	Definitions
Cauchy Sequences Are Bounded	Theorems
Cauchy Sequences Converge	Theorems
Chain Rule	Theorems
Characterization of Compactness in R	Theorems
Characterization of Continuity	Theorems
Closed (Toplogical)	Definitions
Closed iff Convergent Cauchy Sequences	Theorems
Closed Implies Complement is Open (Vice Versa)	Theorems
Closure is the Smallest Closed Set	Theorems
Compact Set (Compactness)	Definitions
Compactness Is Preserved By Continuity	Theorems
Comparison Test (Series)	Theorems
Complement	Definitions
Composition of Continuous Functions	Theorems
Condition for Connected	Theorems
Continuity	Definitions
Continuous Image of Connected Set is Connected	Theorems
Convergence of a Sequence	Definitions
Convergence of a Sequence - Topological Version	Definitions
Countable Sets	Definitions
Darboux's Theorem	Theorems
Dedekind Cuts, Constructing R	8 Additional Topics
Derivative	Definitions
Derivative is Zero Implies Constant Function	Theorems
Differentiability Implies Continuity	Theorems
Dirichlet's Function	Example Functions
Discontinuity (and its types)	Definitions
Divergence	Definitions
Epsilon-Neighborhood	Definitions
Existence of Functional Limits	Theorems
Existence of Square Roots	Theorems
Existence of the Reals	8 Additional Topics
Extreme Value Theorem	Theorems
Field	Definitions
Finite Subcover	Definitions
Functional Limit	Definitions
Functional Limit (Topological Version)	Definitions
Generalized Mean Value Theorem	Theorems
Heine-Borel Theorem (Compact Equivalences)	Theorems
Increasing, Decreasing, Monotone Sequences	Definitions
Infinity	Definitions
Interior Extremum Theorem	Theorems
Intermediate Value Property	Definitions
Intermediate Value Theorem	Theorems
Irrationality of sqrt(2)	Theorems
Isolated Point	Definitions
L'Hopital's Rule(s)	Theorems
Least Upper Bound of R (via Dedekind Cuts)	8 Additional Topics
Least Upper Bounds and Greatest Upper Bounds	Definitions
Limit Laws (Algebraic Limit Theorem)	Theorems
Limit Point	Definitions
Limits and Order (Order Limit Theorem)	Theorems
Limits are Unique	Theorems
Maxima and Minima	Definitions
Mean Value Theorem (MVT)	Theorems
Monotone Convergence Theorem	Theorems
Monotonic Sequences	Definitions
Multiplication on R (via Dedekind Cuts)	8 Additional Topics
Nested Compact Set Property	Theorems
Nested Interval Property	Theorems
One-Sided Limits	Definitions
One-to-One Correspondance	Definitions
Open Cover	Definitions
Open Sets	Definitions
Order	Definitions
Ordered Field	Definitions
Ordering of R (via Dedekind Cuts)	8 Additional Topics
Perfect Sets	Definitions
Perfect Sets are Uncountable	Theorems
Power Set	Definitions
Q (The Rationals)	Important Notes
R - the Reals and Completeness	Definitions
Rearrangement and Rearrangement Theorem	Definitions
Review of Proof Strategies	Important Notes
Review of Set Theory and Functions	Important Notes
Review Sheet - Final Exam	Textbook Notes
Rolle's Theorem (Baby MVT)	Theorems
Separated, Disconnected, Connected Sets	Definitions
Sequence	Definitions
Sequential Criterion for Absence of Uniform Continuity	Theorems
Sequential Criterion For Functional Limits	Theorems
Series, Convergence of A Series	Definitions
Set Addition with the Reals, The Suprema Lemma	Definitions
Subsequence Convergence Theorem	Theorems
Suprema Lemma	Theorems
Test for Divergence (Series)	Theorems
The Cantor Set is Perfect	Theorems
The Density of Q in R	Theorems
Thomae's Function	Example Functions
Triangle Inequality	Theorems
Types of Convergence	Definitions
Uniform Continuity	Definitions
Uniformly Continuous iff Compact and Continuous	Theorems
Union & Intersection of Collections of Open Sets	Theorems
Unions & Intersection of Closed Sets	Theorems
Wierestrass Function	Functions

For problems that I've done:

File	Section, Subsection
53 The Mean Value Theorems	5 The Derivative
52 Derivatives and the Intermediate Value Property	5 The Derivative
45 IVT and Discontinuity Practice	4 Functional Limits and Continuity
44 Uniform Continuity Practice	4 Functional Limits and Continuity
43 Continuity Practice	4 Functional Limits and Continuity
42 Functional Limits Practice	4 Functional Limits and Continuity
34 Perfect and Connected Sets Practice	3 Topology of R
33 Compact Sets Practice	3 Topology of R
32 Open and Closed Sets Practice	3 Topology of R
27 Properties of Infinite Series Practice	2 Sequences and Series
26 Cauchy Sequences Practice	2 Sequences and Series
25 Subsequences and the Bolzano-Weierstrass Theorem Practice	2 Sequences and Series
24 Monotone Sequence, MCT, and Cauchy Condensation Test	2 Sequences and Series
23 Algebraic and Order Limit Theorems Practice	2 Sequences and Series
22 Sequences Practice	2 Sequences and Series
Cardinality Practice	1 The Real Numbers
Cantor's Problems	1 The Real Numbers
More Suprema and Density Practice	1 The Real Numbers
Suprema + Density Practice	1 The Real Numbers
Proof Strategy Practice	1 The Real Numbers
1.3 - Supremum and Infimum	Old Practice
1.2 - Set Theory Review	Old Practice