Sterling K. Berberian

A First Course in Real Analysis (eBook, PDF)

• Format: PDF

Produktbeschreibung

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.

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Produktdetails

• Verlag: Springer US
• Seitenzahl: 240
• Erscheinungstermin: 10. September 2012
• Englisch
• ISBN-13: 9781441985484
• Artikelnr.: 44000378

Inhaltsangabe

1 Axioms for the Field ? of Real Numbers.- 1.1. The field axioms.- 1.2. The order axioms.- 1.3. Bounded sets, LUB and GLB.- 1.4. The completeness axiom (existence of LUB's).- 2 First Properties of ?.- 2.1. Dual of the completeness axiom (existence of GLB's).- 2.2. Archimedean property.- 2.3. Bracket function.- 2.4. Density of the rationals.- 2.5. Monotone sequences.- 2.6. Theorem on nested intervals.- 2.7. Dedekind cut property.- 2.8. Square roots.- 2.9. Absolute value.- 3 Sequences of Real Numbers, Convergence.- 3.1. Bounded sequences.- 3.2. Ultimately, frequently.- 3.3. Null sequences.- 3.4. Convergent sequences.- 3.5. Subsequences, Weierstrass-Bolzano theorem.- 3.6. Cauchy's criterion for convergence.- 3.7. limsup and liminf of a bounded sequence.- 4 Special Subsets of ?.- 4.1. Intervals.- 4.2. Closed sets.- 4.3. Open sets, neighborhoods.- 4.4. Finite and infinite sets.- 4.5. Heine-Borel covering theorem.- 5 Continuity.- 5.1. Functions, direct images, inverse images.- 5.2. Continuity at a point.- 5.3. Algebra of continuity.- 5.4. Continuous functions.- 5.5. One-sided continuity.- 5.6. Composition.- 6 Continuous Functions on an Interval.- 6.1. Intermediate value theorem.- 6.2. n'th roots.- 6.3. Continuous functions on a closed interval.- 6.4. Monotonic continuous functions.- 6.5. Inverse function theorem.- 6.6. Uniform continuity.- 7 Limits of Functions.- 7.1. Deleted neighborhoods.- 7.2. Limits.- 7.3. Limits and continuity.- 7.4. ?,?characterization of limits.- 7.5. Algebra of limits.- 8 Derivatives.- 8.1. Differentiability.- 8.2. Algebra of derivatives.- 8.3. Composition (Chain Rule).- 8.4. Local max and min.- 8.5. Mean value theorem.- 9 Riemann Integral.- 9.1. Upper and lower integrals: the machinery.- 9.2. First properties of upper and lower integrals.- 9.3. Indefinite upper and lower integrals.- 9.4. Riemann-integrable functions.- 9.5. An application: log and exp.- 9.6. Piecewise pleasant functions.- 9.7.Darboux's theorem.- 9.8. The integral as a limit of Riemann sums.- 10 Infinite Series.- 10.1. Infinite series: convergence, divergence.- 10.2. Algebra of convergence.- 10.3. Positive-term series.- 10.4. Absolute convergence.- 11 Beyond the Riemann Integral.- 11.1 Negligible sets.- 11.2 Absolutely continuous functions.- 11.3 The uniqueness theorem.- 11.4 Lebesgue's criterion for Riemann-integrability.- 11.5 Lebesgue-integrable functions.- A.1 Proofs, logical shorthand.- A.2 Set notations.- A.3 Functions.- A.4 Integers.- Index of Notations.