A First Course in Real Analysis
Mathematics is the music of science, and real analysis is the Bach of mathematics. There are many other foolish things I could say about the subject of this book, but the foregoing will give the reader an idea of where my heart lies. The present book was written to support a first course in real ana...
| Main Author: | |
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| Format: | eBook |
| Language: | English |
| Published: |
New York, NY
Springer New York
1994, 1994
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| Edition: | 1st ed. 1994 |
| Series: | Undergraduate Texts in Mathematics
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- §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
- 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