Invitation to real analysis
This book is an introduction to real analysis for a one-semester course aimed at students who have completed the calculus sequence and preferably one other course, such as linear algebra. It does not assume any specific knowledge and starts with all that is needed from sets, logic, and induction. Then there is a careful introduction to the real numbers with an emphasis on developing proof-writing skills. It continues with a logical development of the notions of sequences, open and closed sets (…
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- Full title :Invitation to real analysis / Cesar E. Silva
Summary
This book is an introduction to real analysis for a one-semester course aimed at students who have completed the calculus sequence and preferably one other course, such as linear algebra. It does not assume any specific knowledge and starts with all that is needed from sets, logic, and induction. Then there is a careful introduction to the real numbers with an emphasis on developing proof-writing skills. It continues with a logical development of the notions of sequences, open and closed sets (including compactness and the Cantor set), continuity, differentiation, integration, and series of numbers and functions. A theme in the book is to give more than one proof for interesting facts; this illustrates how different ideas interact and it makes connections among the facts that are being learned. Metric spaces are introduced early in the book, but there are instructions on how to avoid metric spaces for the instructor who wishes to do so. There are questions that check the readers'understanding of the material, with solutions provided at the end. Topics that could be optional or assigned for independent reading include the Cantor function, nowhere differentiable functions, the Gamma function, and the Weierstrass theorem on approximation by continuous functions
Read More ↓Edition Statements
- Publication Date:©2019
Publication Statements
- Publisher(s):American mathematical Society
- Place of Publication:Providence, Rhode Island
- Language:eng
- Publication Date:©2019
The Physical Object
- Description:1 online resource (x, 303 pages)
Edition Identifiers
- ISBN:9781470452339, 1470452332
- oclc:on1104714502
- LC Classification:QA300
- Dewey Decimal:515/.83
Table of Contents
Preliminaries: sets, functions, and induction -- The real numbers and the completeness property -- Sequences -- Topology of the real numbers and metric spaces -- Continuous functions -- Differentiable functions -- Integration -- Series -- Sequences and series of functions
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