An Introduction to Mathematical Analysis Book

Preface     ix
List of Symbols and Notations     xiii
Fundamental Ideas and Assumptions
Introduction     1
Assumptions relating to the field operations     2
Assumptions relating to the ordering of the real numbers     4
Mathematical induction     7
Upper and lower bounds of sets of real numbers     13
Functions     20
Limits and Continuity
Limits of real functions on the positive integers     35
Limits of real functions of a real variable x as x tends to infinity     58
Elementary topological ideas     62
Limits of real functions at finite points     66
Continuity     74
Inverse functions and fractional indices     92
Differentiability
Derivatives     109
General theorems concerning real functions     120
Maxima, minima and convexity     135
Complex numbers and functions     140
Infinite Series
Elementary properties of infinite series     150
Series with non-negative terms     157
Absolute and conditional convergence     166
The decimal notation for real numbers     184
Functions Defined by Power Series
General theory of power series     191
Real power series     204
The exponential and logarithmic functions     209
The trigonometric functions     230
The hyperbolic functions     241
Complex indices     246
Integration
The indefinite integral     260
Interval functions and functions of bounded variation     287
The Riemann-Stieltjes integral     316
The Riemann integral     351
Curves     391
Area     426
Convergence and Uniformity
Upper and lower limits and their applications     437
Further convergence tests for infinite series     452
Uniform convergence     467
Improper integrals     496
Double series     541
Infinite products     563
Hints for Solutions of Exercises     583
Index     601