Lecture Notes and Readings

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

Reading: [JL] Section 0.3

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

Reading: [JL] Section 3.1

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

Reading: [JL] Section 4.3

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

Reading: [JL] Section 6.1

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.

Course Info

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