3. Inequalities and ordered ﬁelds 11

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Exercise 1.13. Type “Non-Archimedean Ordered Field” into an internet search

engine and see what you ﬁnd. Then try to understand one of the examples.

3.3. Limits. Completeness in the sense of Deﬁnition 3.2 (for Archimedean or-

dered ﬁelds) is equivalent to a notion involving limits of Cauchy sequences. See

Remark 3.1. We will carefully discuss these deﬁnitions from a calculus or begin-

ning real analysis course. First we remind the reader of some elementary properties

of the absolute value function. We gain intuition by thinking in terms of distance.

Deﬁnition 3.6. For x ∈ R, we deﬁne |x| by |x| = x if x ≥ 0 and |x| = -x if

x 0. Thus |x| represents the distance between x and 0. In general, we deﬁne the

distance δ(x,y) between real numbers x and y by

δ(x,y) = |x - y|.

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Exercise 1.14. Show that the absolute value function on R satisﬁes the follow-

ing properties:

1) |x| ≥ 0 for all x ∈ R, and |x| = 0 if and only if x = 0.

2) -|x| ≤ x ≤ |x| for all x ∈ R.

3) |x + y| ≤ |x| + |y| for all x,y ∈ R (the triangle inequality).

4) |a - c| ≤ |a - b| + |b - c| for all a,b,c ∈ R (second form of the triangle

inequality).

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Exercise 1.15. Why are properties 3) and 4) of the previous exercise called

triangle inequalities?

We make several comments about Exercise 1.14. First of all, one can prove

property 3) in two rather diﬀerent ways. One way starts with property 2) for x

and y and adds the results. Another way involves squaring. Property 4) is crucial

because of its interpretation in terms of distances. Mathematicians have abstracted

these properties of the absolute value function and introduced the concept of a

metric space. See Section 6.

We recall that a sequence {xn} of real numbers is a function from N to

R. The real number xn is called the n-th term of the sequence. The notation

x1,x2,...,xn,..., where we list the terms of the sequence in order, amounts to list-

ing the values of the function. Thus x : N → R is a function, and we write xn

instead of x(n). The intuition gained from this alteration of notation is especially

valuable when discussing limits.

Deﬁnition 3.7. Let {xn} be a sequence of real numbers. Assume L ∈ R.

• LIMIT. We say that “the limit of xn is L” or that “xn converges to L”, and

we write limn→∞xn = L if the following statement holds: For all 0, there

is an N ∈ N such that n ≥ N implies |xn - L| .

• CAUCHY. We say that {xn} is a Cauchy sequence if the following statement

holds: For all 0, there is an N ∈ N such that m,n ≥ N implies |xm-xn|

.