Part 1 begins with an overview of properties of the real numbers and
starts to introduce the notions of set theory. The absolute value and in
particular inequalities are considered in great detail before functions
and their basic properties are handled. From this the authors move to
differential and integral calculus. Many examples are discussed. Proofs
not depending on a deeper understanding of the completeness of the real
numbers are provided. As a typical calculus module, this part is thought
as an interface from school to university analysis.
Part 2 returns to the structure of the real numbers, most of all to the
problem of their completeness which is discussed in great depth. Once
the completeness of the real line is settled the authors revisit the
main results of Part 1 and provide complete proofs. Moreover they
develop differential and integral calculus on a rigorous basis much
further by discussing uniform convergence and the interchanging of
limits, infinite series (including Taylor series) and infinite products,
improper integrals and the gamma function. In addition they discussed in
more detail as usual monotone and convex functions.
Finally, the authors supply a number of Appendices, among them
Appendices on basic mathematical logic, more on set theory, the Peano
axioms and mathematical induction, and on further discussions of the
completeness of the real numbers.
Remarkably, Volume I contains ca. 360 problems with complete, detailed
solutions.
