A First Course in Real Analysis
The book offers an initiation into mathematical reasoning, and into the mathematician's mind-set and reflexes. Specifically, the fundamental operations of calculus--differentiation and integration of functions and the summation of infinite series--are built, with logical continuity (i.e., "rigor"), starting from the real number system. The first chapter sets down precise axioms for the real number system, from which all else is derived using the logical tools summarized in an Appendix. The disc…
Read More ↓Title Statements
- Full title :A First Course in Real Analysis / by Sterling K. Berberian.
Summary
The book offers an initiation into mathematical reasoning, and into the mathematician's mind-set and reflexes. Specifically, the fundamental operations of calculus--differentiation and integration of functions and the summation of infinite series--are built, with logical continuity (i.e., "rigor"), starting from the real number system. The first chapter sets down precise axioms for the real number system, from which all else is derived using the logical tools summarized in an Appendix. The discussion of the "fundamental theorem of calculus," the focal point of the book, especially thorough. The concluding chapter establishes a significant beachhead in the theory of the Lebesgue integral by elementary means
Read More ↓Edition Statements
- Publication Date:1994.
Publication Statements
- Publisher(s):Springer New York
- Place of Publication:New York, NY
- Language:eng
- Publication Date:1994.
The Physical Object
- Description:1 online resource (xi, 240 pages)
Edition Identifiers
- ISBN:9781441985484, 1441985484
- DOI:10.1007/978-1-4419-8548-4
- oclc:ocn853259755
- LC Classification:QA331.5
- Dewey Decimal:515.8
Table of Contents
1 Axioms for the Field? of Real Numbers -- ʹ1.1. The field axioms -- ʹ1.2. The order axioms -- ʹ1.3. Bounded sets, LUB and GLB -- ʹ1.4. The completeness axiom (existence of LUB's) -- 2 First Properties of? -- ʹ2.1. Dual of the completeness axiom (existence of GLB's) -- ʹ2.2. Archimedean property -- ʹ2.3. Bracket function -- ʹ2.4. Density of the rationals -- ʹ2.5. Monotone sequences -- ʹ2.6. Theorem on nested intervals -- ʹ2.7. Dedekind cut property -- ʹ2.8. Square roots -- ʹ2.9. Absolute value -- 3 Sequences of Real Numbers, Convergence -- ʹ3.1. Bounded sequences -- ʹ3.2. Ultimately, frequently -- ʹ3.3. Null sequences -- ʹ3.4. Convergent sequences -- ʹ3.5. Subsequences, Weierstrass-Bolzano theorem -- ʹ3.6. Cauchy's criterion for convergence -- ʹ3.7. limsup and liminf of a bounded sequence -- 4 Special Subsets of? -- ʹ4.1. Intervals -- ʹ4.2. Closed sets -- ʹ4.3. Open sets, neighborhoods -- ʹ4.4. Finite and infinite sets -- ʹ4.5. Heine-Borel covering theorem -- 5 Continuity -- ʹ5.1. Functions, direct images, inverse images -- ʹ5.2. Continuity at a point -- ʹ5.3. Algebra of continuity -- ʹ5.4. Continuous functions -- ʹ5.5. One-sided continuity -- ʹ5.6. Composition -- 6 Continuous Fun…
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