Basic analysis: Introduction to real analysis

Title Page......Page 1 Notes about these notes......Page 5 About analysis......Page 7 Basic set theory......Page 8 Basic properties......Page 21 The set of real numbers......Page 25 Absolute value......Page 31 Intervals and the size of R......Page 35 Sequences and limits......Page 39 Facts about limits of sequences......Page 47 Limit superior, limit inferior, and Bolzano-Weierstrass......Page 57 Cauchy sequences......Page 65 Series......Page 68 Limits of functions......Page 79 Continuous functions......Page 86 Min-max and intermediate value theorems......Page 92 Uniform continuity......Page 98 The derivative......Page 103 Mean value theorem......Page 109 Taylor's theorem......Page 115 The Riemann integral......Page 117 Properties of the integral......Page 125 Fundamental theorem of calculus......Page 133 Pointwise and uniform convergence......Page 139 Interchange of limits......Page 145 Picard's theorem......Page 151 Further Reading......Page 157 Index......Page 159