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Feb 5, 2010 · An inﬁnite sequence (more brieﬂy, a sequence) of

**real**numbers is a**real**-valued function deﬁned on a set of integers ˚ n ˇ ˇn k. We call the values of the functionthe terms of the sequence. We denote a sequence by listingitsterms inorder; thus, fsng. 1 kDfsk;skC1;:::g: (4.1.1) For example, ˆ 1 n2C1 ˙1 0.Abstract. These are some notes on introductory

**real****analysis**. They cover the properties of the**real**numbers, sequences and series of real numbers, limits of functions, continuity, differentiability, sequences and series of functions, and Riemann integration. They don’t include multi-variable calculus or contain any problem sets.- 2MB

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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...Find the textbook, lecture notes and readings for Real Analysis,

**a course on basic analysis I at MIT.**The notes cover topics such as sets, cardinality, sequences, series, limits, continuity, differentiation and integration.A comprehensive introduction to

**real analysis,**covering set theory, measure theory, integration, differentiation, and harmonic**analysis.**The notes include definitions, examples, exercises, and references for further reading.- 803KB

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This version of Elementary

**Real****Analysis**, Second Edition, is a hypertexted**pdf**ﬁle, suitable for on-screen viewing. For a trade paperback copy of the text, with the same numbering of Theorems and Exercises (but with diﬀerent page numbering), please visit our web site. Direct all correspondence to thomson@sfu.ca.- 5MB

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Abstract. These are some notes on introductory

**real****analysis**. They cover limits of functions, continuity, diﬀerentiability, and sequences and series of functions, but not Riemann integration A background in sequences and series of real numbers and some elementary point set topology of the real numbersPeople also ask

What is introductory real analysis?

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needed to start doing

**real****analysis**. 1.1 Sets, Numbers, and Proofs Let Sbe a set. If xis an element of Sthen we write x∈ S, otherwise we write that x/∈ S. A set Ais called a subset of Sif each element of Ais also an element of S, that is, if a∈ Athen also a∈ S. To denote that Ais a subset of Swe write A⊂ S. Now let Aand Bbe subsets of S.