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.
The lecture notes were prepared by Paige Dote under the guidance of Dr. Rodriguez.
Dr. Rodriguez’s Fall 2020 lecture notes in one file:
Reading: [JL] Section 0.3
- 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.
- 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.
Reading: [JL] Sections 1.1 and 1.2
- 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.
- Fields, ordered fields and examples,
- The fact that the real numbers are characterized as the unique ordered field with the least upper bound property.
Reading: [JL] Sections 1.2, 1.3, 1.5, and 2.1
- The Archimedean property of the real numbers,
- The density of the rational numbers,
- Using sup/inf’s and the absolute value.
- The triangle inequality,
- Decimal representations and the uncountability of the real numbers,
- The definition of sequences of real numbers and convergence of sequences.
Reading: [JL] Sections 2.1 and 2.2
- Monotone sequences and when they have a limit,
- The Squeeze Theorem,
- The relations between limits and order, algebraic operations and the absolute value on the set of real numbers.
Reading: [JL] Sections 2.2, 2.3, 2.4, and 2.5
- The limsup and liminf of a bounded sequence,
- The Bolzano-Weierstrass Theorem.
- Cauchy sequences,
- The definition of convergent series and Cauchy series and some basic properties.
Reading: [JL] Sections 2.5 and 2.6
- Absolute convergence,
- The comparison test,
- The ratio test,
- The root test,
- Alternating series.
Reading: [JL] Section 3.1
- Cluster points,
- Limits of functions,
- The relationship between limits of functions and limits of sequences.
Reading: [JL] Sections 3.1 and 3.2
- The characterization of limits of functions in terms of limits of sequences and applications,
- One-sided limits,
- The definition of continuity.
- 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.
Reading: [JL] Sections 3.3, 3.4, and 4.1
- The min/max theorem for continuous functions on a closed and bounded interval [a,b],
- The bisection method and Bolzano’s intermediate value theorem.
- 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.
Reading: [JL] Sections 4.1 and 4.2
- Differentiability at c implies continuity at c (but not the converse),
- Weierstrass’s construction of a continuous nowhere differentiable function.
- The linearity and various “rules” for the derivative,
- Relative minima and maxima,
- Rolle’s theorem and the mean value theorem.
Reading: [JL] Section 4.3
- Taylor’s theorem,
- Motivation for the Riemann integral,
- Partitions, tags and Riemann sums.
- 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.
Reading: [JL] Section 6.1
- The additive property and inequalities for Riemann integrals,
- The fundamental theorem of calculus,
- Integration by parts and the change of variables formula.
- 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.
Reading: [JL] Sections 6.1 and 6.2
- Uniform convergence implies pointwise convergence for a sequence of functions but not the converse,
- The interchange of limits,
- The Weierstrass M-test.
- Uniform convergence and the interchange of limits for power series,
- The Weierstrass approximation theorem for continuous functions.