In this introductory chapter some mathematical notions are presented rapidly

Definition 1.6 A set A C R admits a maximum if an element XM ^ A
exists such that
X < XM, for any x E A.
The element XM (necessarily unique) is the maximum of the set A and
one denotes it by XM — max^d.
The minimum of a set A, denoted by Xm — min A, is defined in a similar
way.
A set admitting a maximum must be bounded from above: the maximum is an
upper bound for the set, actually the smallest of all possible upper bounds, as we
shall prove. The opposite is not true: a set can be bounded from above but not
admit a maximum, like the set A of (1.6). We know already that 1 is an upper
bound for A. Among all upper bounds, 1 is privileged, being the smallest upper
bound. To convince ourselves of this fact, let us show that each real number r < 1
is not an upper bound, i.e., there is a natural number n such that
n
n + 1 > r.
rr.. 1. . n 7 1 + 1 1 1 1 1 1 - r ^^ , The mequality is equivalent to < - , hence 1 -\— < - , o r — < . This
n r n r n r
r
is to say n > , and the existence of such n follows from property (1.5). So,
1 — r
1 is the smallest upper bound of A., yet not the maximum, for 1 ^ A: there is no
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natural number n such that = 1. One calls 1 the supremum, or least upper
n + 1
bound, of A and writes 1 = sup A.17 bet