358 CHAPTER 15. CONSTRUCTION OF REAL NUMBERS

To show R is complete, let {[x]n}n be a sequence of elements of R. By definition,[x]n = [xn] where xn is a representative of [x]n . Thus xn is in R. Then we are given that forevery ε > 0,ε ∈Q there is nε such that if m,n > nε , then

|[x]n− [x]m| ≡ |[xn]− [xm]| ≡ |[xn−xm]|= [|xn−xm|]< ε

We can therefore, obtain a subsequence, still denoted with n such that∣∣∣[x]n− [x]n+1∣∣∣= [∣∣xn−xn+1∣∣]< 4−n

It will be this subsequence in what follows. Thus there is an increasing sequence {kn} such

that∣∣xn

k − xn+1l

∣∣ < 2−n if k, l ≥ kn where kn is increasing in n. Let y ={

xnkn

}∞

n=1. Then

y ∈ R and if m≥ kn,

|xnm− ym| =

∣∣xnm− xm

km

∣∣≤ ∣∣xnm− xn

kn

∣∣+ ∣∣xnkn− xm

km

∣∣< 2−n +

m−1

∑j=n

∣∣∣x jkn− x j+1

km

∣∣∣< 2−n +∞

∑j=n

2− j = 2−n +2−(n−1)

Then from the above,|[x]n− [y]|= [|xn−y|]< 2−(n−2)

This says that limn→∞ [x]n = [y] by definition. The original Cauchy sequence converges tothe same thing thanks to Theorem 4.5.4.

It follows that you can consider each real number inR as an equivalence class of Cauchysequences. One can show that any two ordered, complete, separable, fields are isomorphicso there is essentially only one of them.

There are other ways to construct the real numbers from the rational numbers. Thetechnique of Dedekind cuts might be a little shorter and easier to understand. However,the above technique of the consideration of equivalence classes of Cauchy sequences canalso be used to complete any metric space and this is a common problem. The techniqueof Dedekind cuts cannot do this because it depends on the order of Q and there is no orderin a general metric space.

A metric space is a nonempty set X on which is defined a distance function (metric)which satisfies the following axioms for x,y,z ∈ X .

d (x,y) = d (y,x) , ∞ > d (x,y)≥ 0

d (x,y)+d (y,z)≥ d (x,z)

d (x,y) = 0 if and only if x = y.

Its completion is a larger metric space with the property that Cauchy sequences con-verge. It will also consist of equivalence classes of Cauchy sequences. The idea of aCauchy sequence makes sense in a metric space.