Roughly speaking， analysis covers more than half of the whole of mathematics.It includes the topics following the limit operation and provides a strong basis for applications of mathematics. Its starting part in the educational process， mathematical analysis， deals with the issues concentrated around continuity.
Roughly speaking， analysis covers more than half of the whole of mathematics.It includes the topics following the limit operation and provides a strong basis for applications of mathematics. Its starting part in the educational process， mathematical analysis， deals with the issues concentrated around continuity.
Many books have been written on mathematical analysis. The list includes a wide spectrum of topics from differential and integral calculus for engineers to the highest mathematical standards such as the books Treatise on Modern Analysis by Dieudonnel and Principles of Mathematical Analysis by Rudin.2 This book occupies a wide range in this spectrum. Its narrower audience includes those who are more or less familiar with differential and integral calculus and would like to get a stronger mathematical background， but the wider audience covers everyone who wants to get rigorous fundamentals in analysis. Therefore， it is suggested as a real analysis textbook for second- or third-year students who have studied differential and integral calculus their first year. At the same time it may serve as a real analysis textbook for first-year students of mathematical departments.
At present， the worldwide number of students is significantly large and is continuously increasing. As a result， the level of an average freshman student is low and is going to get lower. Currently， the classic method of teaching analysis on the rigorous level from the beginning may be acceptable in a restricted number of universities. A large number of universities make a compromise by teaching calculus (i.e.， calculation-based analysis) during freshman year and then set up rigorous mathematical analysis during sophomore or junior years. This is a solution to the problem， but it creates an educational. problem. It is not easy to change the orientation of sophomore or junior students from the problems of calculation nature， which are typical for calculus， to rigorous mathematical analysis. This book aims to serve as a transition from calculus to rigorous analysis.
Preface
1 Sets and Proofs
1．1 Sets， Elements， and Subsets
1．2 Operations on Sets
1．3 Language of Logic
1．4 Techniques of Proof
1．5 Relations
1．6 Functions
1．7 Axioms of Set Theory
Exercises
2 Numbers
2．1 SystemN
2．2 Systems Z and Q
2．3 Least Upper Bound Property and Q
2．4 System R
2．5 Least Upper Bound Property and R
2．6 Systems R， C， and *R
2．7 Cardinality
Exercises
3 Convergence
3．1 Convergence ofNumerical Sequences
3．2 Cauchy Criterion for Convergence
3．3 Ordered Field Structure and Convergence
3．4 Subsequences
3．5 NumericalSeries
3．6 Some Series of Particular Interest
3．7 AbsoluteConvergence
3．8 Number e
Exercises
4 Point Set Topology
4．1 MetricSpaces
4．2 Open and Closed Sets
4．3 Completeness
4．4 Separability
4．5 TotaIBoundedness
4．6 Compactness
4．7 Perfectness
4．8 Connectedness
4．9* Structure of Open and Closed Sets
Exercises
5 Continuity
5．1 Definition and Examples
5．2 Continuity and Limits
5．3 Continuity and Compactness
5．4 Continuity and Connectedness
5．5 Continuity and Oscillation
5．6 Continuity of Rk-valued Functions
Exercises
6 Space C(E， E')
6．1 UniformContinuity
6．2 UniformConvergence
6．3 Completeness of C(E， E)
6．4 Bernstein and Weierstrass Theorems
6．5* Stone and Weierstrass Theorems
6．6* Ascoli-Arzela Theorem
Exercises
7 Differentiation
7．1 Derivative
7．2 Differentiation and Continuity
7．3 Rules of Differentiation
7．4 Mean-ValueTheorems
7．5 Taylor'sTheorem
7．6* DifferentialEquations
7．7* Banach Spaces and the Space C1 (a，b)
7．8 A View to Differentiation in Rk
Exercises
8 Bounded Variation
8．1 Monotone Functions
8．2 CantorFunction
8．3 Functions ofBoundedVariation
8．4 Space BV(a， b)
8．5 Continuous Functions of Bounded Variation
……
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