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  1. 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.

  2. 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.

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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.

  3. 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.

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  4. 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.

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  5. 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 numbers

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  7. 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.

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