Introduction to Mathematical Analysis

Content. - 1. Axioms for the Real Numbers. - 1 Introduction. - 2 Fields. - 3 Order. - 4 Completeness. - 5 Upper bound. - 6 The Archimedean property. - Exercises. - 2. Sequences. - 7 Limit of a sequence. - 8 Sequences without limits. - 9 Monotone sequences. - Exercises. - 3. Series. - 10 Infinite series. - 11 Convergence. - 12 Tests. - 13 Absolute convergence. - 14 Power series. - Exercises. - 4. Continuous Functions. - 15 Limit of a function. - 16 Continuity. - 17 The intermediate value property. - 18 Bounds of a continuous function. - Exercises. - 5. Differentiable Functions. - 19 Derivatives. - 20 Rolle's theorem. - 21 The mean value theorem. - Exercises. - 6. The Riemann Integral. - 22 Introduction. - 23 Upper and lower sums. - 24 Riemann-integrable functions. - 25 Examples. - 26 A necessary and sufficient condition. - 27 Monotone functions. - 28 Uniform continuity. - 29 Integrability of continuous functions. - 30 Properties of the Riemann integral. - 31 The mean value theorem. - 32 Integration and differentiation. - Exercises. - Answers to the Exercises. Language: English