A Sequential Introduction to Real Analysis

Description

Real analysis provides the fundamental underpinnings for calculus, arguably the most useful and influential mathematical idea ever invented. It is a core subject in any mathematics degree, and also one which many students find challenging. A Sequential Introduction to Real Analysis gives a fresh take on real analysis by formulating all the underlying concepts in terms of convergence of sequences. The result is a coherent, mathematically rigorous, but conceptually simple development of the standard theory of differential and integral calculus ideally suited to undergraduate students learning real analysis for the first time. This book can be used as the basis of an undergraduate real analysis course, or used as further reading material to give an alternative perspective within a conventional real analysis course. Readership: Undergraduate mathematics students taking a course in real analysis Content: Machine generated contents note: ch. 1 Basic properties of the set of real numbers -- 1.1. Recap of set notation -- 1.2. Functions -- 1.3. Boundedness and the Axiom of Completeness -- 1.4. Some consequences of the Axiom of Completeness -- 1.5. Summary -- 1.6. Tutorial problems -- 1.7. Homework problems -- ch. 2 Real sequences -- 2.1. Definition and examples of real sequences -- 2.2. Convergence of a real sequence -- 2.3. Summary -- 2.4. Tutorial problems -- 2.5. Homework problems -- ch. 3 Limit theorems -- 3.1. Some basic limit theorems -- 3.2. The Monotone Convergence Theorem -- 3.3. Sequences and suprema -- 3.4. Summary -- 3.5. Tutorial problems -- 3.6. Homework problems -- ch. 4 Subsequences -- 4.1. Definition and convergence properties -- 4.2. The Bolzano Weierstrass Theorem -- 4.3. Summary -- 4.4. Tutorial problems -- 4.5. Homework problems -- ch. 5 Series -- 5.1. Definition and convergence -- 5.2. Convergence tests for series -- 5.3. Alternating series -- 5.4. Absolute convergence -- 5.5. Summary. Note continued: 5.6. Tutorial problems -- 5.7. Homework problems -- ch. 6 Continuous functions -- 6.1. Sequential continuity -- 6.2. Basic properties of continuous functions -- 6.3. The Intermediate Value Theorem -- 6.4. The Extreme Value Theorem -- 6.5. Summary -- 6.6. Tutorial problems -- 6.7. Homework problems -- ch. 7 Some symbolic logic -- 7.1. Statements and their symbolic manipulation -- 7.2. Implications -- 7.3. Quantifiers -- 7.4. Summary -- 7.5. Tutorial problems -- 7.6. Homework problems -- ch. 8 Limits of functions -- 8.1. The main definition -- 8.2. Limits at infinity -- 8.3. Summary -- 8.4. Tutorial problems -- 8.5. Homework problems -- ch. 9 Differentiable functions -- 9.1. The main definition -- 9.2. The rules of differentiation -- 9.3. Functions differentiable on an interval -- 9.4. Higher derivatives and Taylor's Theorem -- 9.5. Summary -- 9.6. Tutorial problems -- 9.7. Homework problems -- ch. 10 Power series -- 10.1. Definition and radius of convergence. Note continued: 10.2. Differentiability of power series -- 10.3. Properties of the exponential function -- 10.4. Elementary properties of the trigonometric functions -- 10.5. Summary -- 10.6. Tutorial problems -- 10.7. Homework problems -- ch. 11 Integration -- 11.1. Dissections and Riemann sums -- 11.2. Definition of the Riemann integral -- 11.3.A sequential characterization of integrability -- 11.4. Elementary properties of the Riemann integral -- 11.5. The Fundamental Theorem of the Calculus -- 11.6. Summary -- 11.7. Tutorial problems -- 11.8. Homework problems -- ch. 12 Logarithms and irrational powers -- 12.1. Logarithms -- 12.2. Irrational (and rational) powers -- 12.3. Summary -- 12.4. Tutorial problems -- 12.5. Homework problems -- ch. 13 What are the reals? -- 13.1. What arc the rationals? -- 13.2. The Cauchy property -- 13.3.A sequential construction of the reals.