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 function 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 M0). 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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