A First Course in Real Analysis

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.