2 1. Arithmetic of the p-adic Numbers
1.1. From Q to R; the concept of completion
The real numbers, denoted by R, are obtained from the rationals by
a procedure called completion. This procedure can be applied to any
metric space, i.e., to a space M with a metric d on it. Recall that a
d: M x M -*m
defined on all ordered pairs (a?, y) of elements of a nonempty set M
is said to be a metric if it possesses the following properties:
(1) d(x, y) 0; d(x, y) = 0 if and only if x = y;
(2) d(xiy) = d(y}x) y x.yeM;
(3) d(x, y) d(x} z) + d(z, y)V x,y,ze M.
The function d is also called the distance function.
We say that a sequence {rn} in a metric space M is a Cauchy
sequence if for any e 0 there exists a positive integer N such that
n,m N implies d(rn, rm) e. If any Cauchy sequence in M has a
limit in M, then M is called a complete metric space.
Theorem 1.1 (Completion Theorem). Every metric space M can be
completed, i.e., there exists a metric space (M,D) such that
(1) M is complete with respect to the metric D;
(2) M contains a subset Mo isometric to M;
(3) Mo is dense in M (i.e., each point in M is a limit point for
The proof that can be found e.g. in [13, Theorem 76] consists
in an explicit construction of the completion M and the metric D
on it. We start with the collection {M} of all Cauchy sequences in
My convergent or not, and turn it into a metric space. But first we
introduce an equivalence relation on {M}: two Cauchy sequences an
and bn are called equivalent, we write {an} ~ {6n}, if d(anibn) 0.
(It is easy to check that this is an equivalence relation on {M}.)
We define M to be the set of equivalence classes, M = {M}/ ~.
The metric D between two equivalence classes of Cauchy sequences
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