# Real Analysis

### Textbook

Lebl, Jiří. Basic Analysis I: Introduction to Real Analysis, Volume 1. CreateSpace Independent Publishing Platform, 2018. ISBN: 9781718862401.

[JL] = Basic Analysis: Introduction to Real Analysis (Vol. 1) (PDF - 2.2MB) by Jiří Lebl, June 2021 (used with permission)

This book is available as a free PDF download. You can purchase a paper copy by following a link at the same site.

### Lecture Notes

The lecture notes were prepared by Paige Dote under the guidance of Dr. Rodriguez.

Dr. Rodriguez’s Fall 2020 lecture notes in one file:

### Week 1

Lecture 1: Sets, Set Operations, and Mathematical Induction (PDF)

Lecture 1: Sets, Set Operations, and Mathematical Induction (TEX)

• Sets and their operations (union, intersection, complement, DeMorgan’s laws),
• The well-ordering principle of the natural numbers,
• The theorem of mathematical induction and applications.

Lecture 2: Cantor’s Theory of Cardinality (Size) (PDF)

Lecture 2: Cantor’s Theory of Cardinality (Size) (TEX)

• Injective, surjective and bijective functions,
• Cantor’s theory of the cardinality (size) of sets,
• Countable sets and the cardinality of the power set compared to the cardinality of the original set.

### Week 2

Reading: [JL] Sections 1.1 and 1.2

Lecture 3: Cantor’s Remarkable Theorem and the Rationals’ Lack of the Least Upper Bound Property (PDF)

Lecture 3: Cantor’s Remarkable Theorem and the Rationals’ Lack of the Least Upper Bound Property (TEX)

• Cantor’s theorem about the cardinality of the power set of a set,
• Ordered sets and the least upper bound property,
• The fact that the rationals do not have the least upper bound property.

Lecture 4: The Characterization of the Real Numbers (PDF)

Lecture 4: The Characterization of the Real Numbers (TEX)

• Fields, ordered fields and examples,
• The fact that the real numbers are characterized as the unique ordered field with the least upper bound property.

### Week 3

Reading: [JL] Sections 1.2, 1.3, 1.5, and 2.1

Lecture 5: The Archimedian Property, Density of the Rationals, and Absolute Value (PDF)

Lecture 5: The Archimedian Property, Density of the Rationals, and Absolute Value (TEX)

• The Archimedean property of the real numbers,
• The density of the rational numbers,
• Using sup/inf’s and the absolute value.

Lecture 6: The Uncountabality of the Real Numbers (PDF)

Lecture 6: The Uncountabality of the Real Numbers (TEX)

• The triangle inequality,
• Decimal representations and the uncountability of the real numbers,
• The definition of sequences of real numbers and convergence of sequences.

### Week 4

Reading: [JL] Sections 2.1 and 2.2

Lecture 7: Convergent Sequences of Real Numbers (PDF)

Lecture 7: Convergent Sequences of Real Numbers (TEX)

• Monotone sequences and when they have a limit,
• Subsequences.

Lecture 8: The Squeeze Theorem and Operations Involving Convergent Sequences (PDF)

Lecture 8: The Squeeze Theorem and Operations Involving Convergent Sequences (TEX)

• The Squeeze Theorem,
• The relations between limits and order, algebraic operations and the absolute value on the set of real numbers.

### Week 5

Reading: [JL] Sections 2.2, 2.3, 2.4, and 2.5

Lecture 9: Limsup, Liminf, and the Bolzano-Weierstrass Theorem (PDF)

Lecture 9: Limsup, Liminf, and the Bolzano-Weierstrass Theorem (TEX)

• The limsup and liminf of a bounded sequence,
• The Bolzano-Weierstrass Theorem.

Lecture 10: The Completeness of the Real Numbers and Basic Properties of Infinite Series (PDF)

Lecture 10: The Completeness of the Real Numbers and Basic Properties of Infinite Series (TEX)

• Cauchy sequences,
• The definition of convergent series and Cauchy series and some basic properties.

### Week 6

Reading: [JL] Sections 2.5 and 2.6

Lecture 11: Absolute Convergence and the Comparison Test for Series (PDF)

Lecture 11: Absolute Convergence and the Comparison Test for Series (TEX)

• Absolute convergence,
• The comparison test,
• p-series.

Lecture 12: The Ratio, Root, and Alternating Series Tests (PDF)

Lecture 12: The Ratio, Root, and Alternating Series Tests (TEX)

• The ratio test,
• The root test,
• Alternating series.

### Week 7

Lecture 13: Limits of Functions (PDF)

Lecture 13: Limits of Functions (TEX)

• Cluster points,
• Limits of functions,
• The relationship between limits of functions and limits of sequences.

### Week 8

Reading: [JL] Sections 3.1 and 3.2

Lecture 14: Limits of Functions in Terms of Sequences and Continuity (PDF)

Lecture 14: Limits of Functions in Terms of Sequences and Continuity (TEX)

• The characterization of limits of functions in terms of limits of sequences and applications,
• One-sided limits,
• The definition of continuity.

Lecture 15: The Continuity of Sine and Cosine and the Many Discontinuities of Dirichlet’s Function (PDF)

Lecture 15: The Continuity of Sine and Cosine and the Many Discontinuities of Dirichlet’s Function (TEX)

• The characterization of continuity in terms of limits of sequences and applications,
• The continuity of sin(x) and cos(x),
• A function which is discontinuous at every point of the real number line.

### Week 9

Reading: [JL] Sections 3.3, 3.4, and 4.1

Lecture 16: The Min/Max Theorem and Bolzano’s Intermediate Value Theorem (PDF)

Lecture 16: The Min/Max Theorem and Bolzano’s Intermediate Value Theorem (TEX)

• The min/max theorem for continuous functions on a closed and bounded interval [a,b],
• The bisection method and Bolzano’s intermediate value theorem.

Lecture 17: Uniform Continuity and the Definition of the Derivative (PDF)

Lecture 17: Uniform Continuity and the Definition of the Derivative (TEX)

• The definition of uniform continuity,
• The equivalence of continuity and uniform continuity for functions on a closed and bounded interval [a,b],
• The definition of the derivative.

### Week 10

Reading: [JL] Sections 4.1 and 4.2

Lecture 18: Weierstrass’s Example of a Continuous and Nowhere Differentiable Function (PDF)

Lecture 18: Weierstrass’s Example of a Continuous and Nowhere Differentiable Function (TEX)

• Differentiability at c implies continuity at c (but not the converse),
• Weierstrass’s construction of a continuous nowhere differentiable function.

Lecture 19: Differentiation Rules, Rolle’s Theorem, and the Mean Value Theorem (PDF)

Lecture 19: Differentiation Rules, Rolle’s Theorem, and the Mean Value Theorem (TEX)

• The linearity and various “rules” for the derivative,
• Relative minima and maxima,
• Rolle’s theorem and the mean value theorem.

### Week 11

Lecture 20: Taylor’s Theorem and the Definition of Riemann Sums (PDF)

Lecture 20: Taylor’s Theorem and the Definition of Riemann Sums (TEX)

• Taylor’s theorem,
• Motivation for the Riemann integral,
• Partitions, tags and Riemann sums.

Lecture 21: The Riemann Integral of a Continuous Function (PDF)

Lecture 21: The Riemann Integral of a Continuous Function (TEX)

• The definition and proof of existence of the Riemann integral for a continuous function on a closed and bounded interval,
• The linearity of the Riemann interval.

### Week 12

Lecture 22: The Fundamental Theorem of Calculus, Integration by Parts, and Change of Variable Formula (PDF)

Lecture 22: The Fundamental Theorem of Calculus, Integration by Parts, and Change of Variable Formula (TEX)

• The additive property and inequalities for Riemann integrals,
• The fundamental theorem of calculus,
• Integration by parts and the change of variables formula.

Lecture 23: Pointwise and Uniform Convergence of Sequences of Functions (PDF)

Lecture 23: Pointwise and Uniform Convergence of Sequences of Functions (TEX)

• The Riemann-Lebesgue lemma for Fourier coefficients (as an application of integration by parts),
• The definitions of pointwise convergence and uniform convergence of sequences of functions.

### Week 13

Reading: [JL] Sections 6.1 and 6.2

Lecture 24: Uniform Convergence, the Weierstrass M-Test, and Interchanging Limits (PDF)

Lecture 24: Uniform Convergence, the Weierstrass M-Test, and Interchanging Limits (TEX)

• Uniform convergence implies pointwise convergence for a sequence of functions but not the converse,
• The interchange of limits,
• The Weierstrass M-test.

Lecture 25: Power Series and the Weierstrass Approximation Theorem (PDF)

Lecture 25: Power Series and the Weierstrass Approximation Theorem (TEX)

• Uniform convergence and the interchange of limits for power series,
• The Weierstrass approximation theorem for continuous functions.

Fall 2020
Lecture Notes
Lecture Videos
Exams
Problem Sets