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Chapter 1 Introduction. 1.1 Preliminary Remarks. These Notes provide an introduction to 20th century mathematics, and in particular to Mathematical Analysis, which roughly speaking is the \in depth" study of Calculus. All of the Analysis material from B21H and some of the material from B30H is included here.
The lecture notes contain topics of real analysis usually covered in a 10-week course: the completeness axiom, sequences and convergence, continuity, and differentiation. The lecture notes also contain many well-selected exercises of various levels.
9 de dic. de 2019 · Mathematical analysis is a cornerstone of mathematics. As such, the content of this book is highly relevant to any mathematical scientist. The text provides a solid foundation for students of mathematics, physics, chemistry, or engineering.
- Elias Zakon
5 de feb. de 2010 · SECTION 1.2 emphasizes the principleof mathematical induction. SECTION 1.3 introduces basic ideas of set theory in the context of sets of real num-bers. In this section we prove two fundamental theorems: the Heine–Borel and Bolzano– Weierstrass theorems. 1.1 THE REAL NUMBER SYSTEM
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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 follo...
The lecture notes were prepared by Paige Dote under the guidance of Dr. Rodriguez. Dr. Rodriguez’s Fall 2020 lecture notes in one file: 1. Real Analysis (PDF) 2. Real Analysis (ZIP)LaTeX source files
Reading: [JL] Section 0.3 Lecture 1: Sets, Set Operations, and Mathematical Induction (PDF) Lecture 1: Sets, Set Operations, and Mathematical Induction (TEX) 1. Sets and their operations (union, intersection, complement, DeMorgan’s laws), 2. The well-ordering principle of the natural numbers, 3. The theorem of mathematical induction and application...
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) 1. Cantor’s theorem about the cardinality of the power set of a set, 2. Ordered sets and the least upper boun...
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) 1. The Archimedean property of the real numbers, 2. The density of the rational numbers, 3. Using sup/inf’s and the absolute val...
Reading: [JL] Sections 2.1 and 2.2 Lecture 7: Convergent Sequences of Real Numbers (PDF) Lecture 7: Convergent Sequences of Real Numbers (TEX) 1. Monotone sequences and when they have a limit, 2. Subsequences. Lecture 8: The Squeeze Theorem and Operations Involving Convergent Sequences (PDF) Lecture 8: The Squeeze Theorem and Operations Involving C...
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) 1. The limsup and liminf of a bounded sequence, 2. The Bolzano-Weierstrass Theorem. Lecture 10: The Completeness of the Real Numbers and Basic Properties of Infinite ...
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) 1. Absolute convergence, 2. The comparison test, 3. p-series. Lecture 12: The Ratio, Root, and Alternating Series Tests (PDF) Lecture 12: The Ratio, Root, and Alternat...
Reading: [JL] Section 3.1 Lecture 13: Limits of Functions (PDF) Lecture 13: Limits of Functions (TEX) 1. Cluster points, 2. Limits of functions, 3. The relationship between limits of functions and limits of sequences.
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) 1. The characterization of limits of functions in terms of limits of sequences and applications, 2. One-sided limits, 3. The definition of continuity. Lecture 15: Th...
Let us give the most important difference between analysis and algebra. In algebra, we prove equalities directly; we prove that an object, a number perhaps, is equal to another object. In analysis, we usually prove inequalities, and we prove those inequalities by estimating. To illustrate the point, consider the following statement.
An Introduction to Mathematical Analysis. by Malcolm R. Adams. c 2007. Chapter 1. Sequences. 1.1 The general concept of a sequence. We begin by discussing the concept of a sequence. Intuitively, a sequence is an ordered list of objects or events.
