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|
.
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