A First Course in Real Analysis (Undergraduate Texts in Mathematics)

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 analysis, normally taken after a year of elementary calculus. Real analysis is, roughly speaking, the modern setting for Calculus, “real” alluding to the field of real numbers that underlies it all. At center stage are functions, defined and taking values in sets of real numbers or in sets (the plane, 3-space, etc.) readily derived from the real numbers; a first course in real analysis traditionally places the emphasis on real-valued functions defined on sets of real numbers. The agenda for the course: (1) start with the axioms for the field ofreal numbers, (2) build, in one semester and with appropriate rigor, the foun dations of calculus (including the “Fundamental Theorem”), and, along the way, (3) develop those skills and attitudes that enable us to continue learning mathematics on our own. Three decades of experience with the exercise have not diminished my astonishment that it can be done. 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.