§l. Convergent sequences and series
61
In the language of decimal approximations: there exists an n such that the
property P(x) is true provided that the first n decimal places of x agree with
those of a. The equivalence with (1) can be seen from the observation that,
first, the powers of 10 are among the numbers r > 0, secondly, that if (1)
is satisfied for an r which is not a power of 10, it will be satisfied a fortiori
if one replaces r by a number 1O-n with n sufficiently large that 1O-n < r.
From time to time we will give translations into this language.
The fundamental point to remember in these modes of expression is that,
if one has a finite number of assertions PI (x), ... , Pn (x), and if each of these
assertions, taken separately, is true on a neighbourhood of a given point a, or
for x sufficiently large, then so is the logical conjunction of the given relations;
in other words, they are simultaneously true on a neighbourhood of a, or for
x sufficiently large. Indeed, in the second case there are numbers AI,· .. ,An
such that Pi (x) is true for all x > Ai, so that the n assertions considered will
simultaneously be true when x exceeds the largest of the numbers Ai. In the
first case, there are open balls B(a, rl), ... , B(a, rn) with centre a in which
the corresponding assertions are valid; they will thus be simultaneously valid
in the intersection of these balls, which is the open ball B(a, r) whose radius
is the least of the radii ri of the balls considered.
Contrariwise, the intersection of infinitely many open balls with centre a
(resp. of infinitely many intervals of the form JA, +oo[) can very well reduce
to a point a (resp. be empty): this is the case of the balls B (0, 1 In), n EN, by
Archimedes' axiom. We shall prove later that, in JR., every intersection of intervals is again an interval, possibly empty, but the intersection of an infinite
number of open intervals need not again be an open interval; the intervals
] - lin, l/n[ provide a counterexample.
These concepts are particularly useful when one wants to compare the
"orders of magnitude" - a naIve expression having no mathematically precise
meaning l l - oftwo scalar functions f(x) and g(x) when the variable increases
11 This is not the same as in physics, where an "order of magnitude" means a
factor 10. Example: the power of an atomic bomb is "three orders of magnitude"
(Le. 10 x 10 x 10 = 10 3 ) greater than that of a classical explosive. One of the
experts in the subject even claims that the prestige attached to the megatonne,
or to a million dollars, is linked to the fact that men have ten fingers; Herbert
York, Race to Oblivion (Simon & Schuster, 1970), pp. 89-90: "We picked a onemegaton yield for the Atlas warhead for the same reason that everyone speaks of
rich men as being millionaires and never as being tenmillionaires or one-hundredthousandaires. It really was that mystical, and I was one of the mystics. Thus,
the actual physical size of the first Atlas warhead and the number of people
it would kill were determined by the fact that human beings have two hands
with five fingers each and therefore count by tens". The committee entrusted
with deciding the characteristics of the Atlas in 1953-1955 was chaired by J. von
Neumann. York's book is a sparkling exposition of the American contribution to
the arms race before 1970.
61
In the language of decimal approximations: there exists an n such that the
property P(x) is true provided that the first n decimal places of x agree with
those of a. The equivalence with (1) can be seen from the observation that,
first, the powers of 10 are among the numbers r > 0, secondly, that if (1)
is satisfied for an r which is not a power of 10, it will be satisfied a fortiori
if one replaces r by a number 1O-n with n sufficiently large that 1O-n < r.
From time to time we will give translations into this language.
The fundamental point to remember in these modes of expression is that,
if one has a finite number of assertions PI (x), ... , Pn (x), and if each of these
assertions, taken separately, is true on a neighbourhood of a given point a, or
for x sufficiently large, then so is the logical conjunction of the given relations;
in other words, they are simultaneously true on a neighbourhood of a, or for
x sufficiently large. Indeed, in the second case there are numbers AI,· .. ,An
such that Pi (x) is true for all x > Ai, so that the n assertions considered will
simultaneously be true when x exceeds the largest of the numbers Ai. In the
first case, there are open balls B(a, rl), ... , B(a, rn) with centre a in which
the corresponding assertions are valid; they will thus be simultaneously valid
in the intersection of these balls, which is the open ball B(a, r) whose radius
is the least of the radii ri of the balls considered.
Contrariwise, the intersection of infinitely many open balls with centre a
(resp. of infinitely many intervals of the form JA, +oo[) can very well reduce
to a point a (resp. be empty): this is the case of the balls B (0, 1 In), n EN, by
Archimedes' axiom. We shall prove later that, in JR., every intersection of intervals is again an interval, possibly empty, but the intersection of an infinite
number of open intervals need not again be an open interval; the intervals
] - lin, l/n[ provide a counterexample.
These concepts are particularly useful when one wants to compare the
"orders of magnitude" - a naIve expression having no mathematically precise
meaning l l - oftwo scalar functions f(x) and g(x) when the variable increases
11 This is not the same as in physics, where an "order of magnitude" means a
factor 10. Example: the power of an atomic bomb is "three orders of magnitude"
(Le. 10 x 10 x 10 = 10 3 ) greater than that of a classical explosive. One of the
experts in the subject even claims that the prestige attached to the megatonne,
or to a million dollars, is linked to the fact that men have ten fingers; Herbert
York, Race to Oblivion (Simon & Schuster, 1970), pp. 89-90: "We picked a onemegaton yield for the Atlas warhead for the same reason that everyone speaks of
rich men as being millionaires and never as being tenmillionaires or one-hundredthousandaires. It really was that mystical, and I was one of the mystics. Thus,
the actual physical size of the first Atlas warhead and the number of people
it would kill were determined by the fact that human beings have two hands
with five fingers each and therefore count by tens". The committee entrusted
with deciding the characteristics of the Atlas in 1953-1955 was chaired by J. von
Neumann. York's book is a sparkling exposition of the American contribution to
the arms race before 1970.