This book is a result of the MASS course "Real and p-adic analysis"
that I gave in the MASS program in the fall of 2000. The notes were
first published in MASS Selecta [12], and a Russian translation of a
revised version appeared in [7]. The present text is further revised
and expanded.
The choice of the topic was motivated by the internal beauty of
the subject of p-adic analysis, an unusual one in the undergraduate
curriculum, and abundant opportunities to compare it with its much
more familiar real counterpart.
There are several pedagogical advantages of this approach. Both
real and p-adic numbers are obtained from the rationals by a proce-
dure called completion, which can be applied to any metric space, by
using different distances on the rationals: the usual Euclidean dis-
tance for the reals and a new p-adic distance for each prime p, for the
p-adics. The p-adic distance satisfies the "strong triangle inequality"
that causes surprising properties of p-adic numbers and leads to in-
teresting deviations from the classical real analysis much like how the
renunciation of the fifth postulate of Euclid's Elements, the axiom of
parallels, leads to non-Euclidean geometry. Similarities, on the other
hand, arise when the fact does not depend on the "strong triangle
inequality", and in these cases the same proof works in the real and
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