Advanced calculus of a single variable

摘要

The book starts in a fairly standard way with the notation and terminology associated with sets, functions, logic, intervals, order and inequalities. Then the detailed work begins with limits of sequences and the algebra of limits. The role of the innocuous-looking Archimedean property is highlighted and appealed to at various points. Then comes the choice of how to characterise the real numbers and distinguish them from the rationals. This book goes for the slightly unfashionable choice of the Cauchy convergence criterion, justified by the fact that decimals are Cauchy sequences, but then shows carefully how this leads to both the nestedinterval criterion and the least upper-bound principal. Other things proved with great thoroughness and attention to details are that the irrationals are uncountable and dense in , and that every real number can be represented by a decimal. It stops short of actually constructing the reals as equivalence classes of Cauchy sequences, but directs readers to a source for this. The chapter concludes with cautions about manipulating the ∞ symbol.