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