I have tried to provide an introduction, at an elementary level, to
some of the important topics in real analysis, without avoiding
reference to the central role which the completeness of the real
numbers plays throughout. Many elementary textbooks are written on
the assumption that an appeal to the complete ness axiom is beyond
their scope; my aim here has been to give an account of the
development from axiomatic beginnings, without gaps, while keeping
the treatment reasonably simple. Little previous knowledge is
assumed, though it is likely that any reader will have had some
experience of calculus. I hope that the book will give the
non-specialist, who may have considerable facility in techniques,
an appreciation of the foundations and rigorous framework of the
mathematics that he uses in its applications; while, for the
intending mathe matician, it will be more of a beginner's book in
preparation for more advanced study of analysis. I should finally
like to record my thanks to Professor Ledermann for the suggestions
and comments that he made after reading the first draft of the
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