By Ionin Y. J., Shrikhande M. S.

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1 Let Ik be a field and d(x, y) = Ix - yl· The function d(x, y) is called the metric induced by the absolute value. The definition of d(x, y) parallels, of course, the usual way we define the distance between two real numbers. The first point we need to make is that a great many of the notions that we can define using the usual distance on ffi. work just as well for any old distance. Problem 43 Show that d(x, y) has the following properties: i) for any x, y E lk, d(x, y) ~ ii) for any x, y Elk, d(x,y) = d(y,x) 0, and d(x, y) = 0 if and only if x = y iii) for any x, y, z E lk, d(x,z):::: d(x,y) +d(y,z) These are the general defining properties for a metric; the last inequality is called the triangle inequality, since it expresses the usual fact that the sum of the lengths of two legs of a triangle is bigger than the length of the other side.

Problem 72 Show that the condition lim IX n+l - xnl n_oo =0 is not the same as the Cauchy condition, by showing that there exists a sequence of real numbers that satisfies this condition but is not a Cauchy sequence. In informal terms, the Cauchy condition is stronger than the assertion that successive terms of the sequence get closer and closer together. (Hint: one example of such a sequence was met in Calculus, in the portion on series ... ) Our reason for recalling these notions is that, as our theory now stands, the archimedian absolute value 1 100 is different from all the rest, because there exists an inclusion Q ~ ~ of Q into a field ~ (yes, we do mean the real numbers) which satisfies the following conditions: • the absolute value 1100 extends to • ~ ~, is complete with respect to the metric given by this absolute value, and • Q is dense in ~ (with respect to the metric given by 1100)' This is all probably well-known to the reader (see the standard references for proofs).

To do that, we begin with the set of all Cauchy sequences as the basic object, then use the algebraic operations on Q to handle the resulting object. 4 Let I I = I Ip be a non-archimedian absolute value on Q. We denote bye, or ep(Q) if we want to emphasize p and Q, the set of all Cauchy sequences of elements of Q: e = ep(Q) = {(xn) : (xn) is a Cauchy sequence with respect to lip}. The first thing to check is that e has a natural ring structure, using the "obvious" definitions for the sum and product of two sequences.