Page 20 - Becoming Metric Wise

9
Introduction

the property that consecutive terms of the sequence become arbitrarily
close to each other necessarily has the property that the terms in this
sequence become arbitrarily close to some specific real number. In
mathematical terminology, this means that the set of real numbers is
complete.
For simplicity, and following famous mathematicians such as Donald
Knuth (the inventor of TeX, the standard typesetting system for mathe-
matical texts) we will further on only use the bold face notation for num-
bers, e.g., N and not the so-called blackboard bold, e.g., N in which
certain lines of the symbol are doubled.

1.5.2 Sequences
We have already used the term sequence, but not yet defined it formally.
A sequence is an ordered collection of objects. It should be observed that
repetitions are allowed. Like a set, a sequence contains members, usually
referred to as terms. An example of a sequence is u 5 (1, 5,12, 12,17, 20).
The number of elements (possibly infinite) is called the length of the
sequence. The length of the above sequence u is 6. Formally, a sequence can
be defined as a function whose domain is the natural numbers, with or
without including zero. When the domain consists of the natural numbers
the sequence is by definition infinitely long. In practical applications one
may identify a sequence ending with infinitely many zeros with the finite
sequence ending with the last nonzero element. The all-zero sequence is
then not considered to be a finite sequence. Sequences are denoted as:
u: nAN-u n AR or ðu n Þ in short:
n
A sequence is increasing if for all n: u n # u n11 . It is strictly increasing
if for all n: u n , u n11 . Similarly, a sequence is decreasing if for all
n: u n $ u n11 and strictly decreasing if for all n: u n . u n11 .
A constant sequence is decreasing as well as increasing. The example
sequence u is increasing but not strictly increasing.

1.5.3 Sets
Following Halmos (1960) we simply say that to define a set is to deter-
mine its members. In other words, a set X is determined if and only if
one can tell whether or not any given object x belongs to X. Members
of a set X are often characterized by possessing a common property. For
example, one can consider the set of all prime numbers. More formally