§l. Convergent sequences and series
57
and deduce that
for all complex x and y: if you want to calculate a sum of 100 numbers to
within 0.1, it is prudent to calculate each term to within 0.001.
If one defines the distance between two complex (or real!) numbers x and
y by the formula
d(x,y) = Ix - yl
whose geometric origin is quite clear, the triangle inequality amounts to saying that
(2.1')
d(x, y) ::; d(x, z) + d(z, y)
for all x, y, z E IC. We shall often use the notation d( a, b) to prepare the reader
for the extensions of analysis to functions of several variables, i.e. defined on
a subset of a real vector space of finite dimension such as lR P , or to much
more general spaces (Appendix to Chap. III).
Archimedes' axiom might appear obvious: one can, like John D. Rockefeller, amass a billion 1910 dollars, at five grams of gold to the dollar, by
saving one dollar a day for a sufficiently long time. But this does not follow
from (I) and (II): the professional mathematicians, in general more competent than the amateurs as in all the sports, have long since invented strange
''totally ordered nonarchimedean fields" which satisfy (I) and (II) but not
(III). One does not meet them in our usual mathematics.
One of the consequences of (III) is that if a real number x satisfies x ::; y
for all strictly positive y, then x ::; 0, for if not there would exist an integer
n such that nx > 1, whence x> y with y = lin> O. Other formulations:
(III') for all a, b E lR with a < b, there exists a u E Q such that a < u < b.
(III") for all a E lR and r > 0, there exists an x E Q such that dCa, x) < r.
Let us first prove (III"), which is obvious if one accepts the decimal expression for real numbers. If not, one proceeds as follows. The late lamented
Archimedes provides us an integer p > 0 such that lip < r; one can thus
restrict to the case where r = lip. The inequality to resolve can be written
lpa - pxl < 1, i.e.
b -1 < y < b + 1,
where b = pa and where y = px is neither more nor less rational than x. We
shall even show that, in this case, one can choose y in Z. If b > 0, there are
integers n such that b < n; the least of these then satisfies n - 1 < b < n,
whence b - 1 < n < b + 1 as desired. If b = 0, one takes y = 0. If b < 0, one
reduces to resolving b' -1 < y' < b' + Ion putting b' = -b > ° and y' = -yo
Whence (III").
57
and deduce that
for all complex x and y: if you want to calculate a sum of 100 numbers to
within 0.1, it is prudent to calculate each term to within 0.001.
If one defines the distance between two complex (or real!) numbers x and
y by the formula
d(x,y) = Ix - yl
whose geometric origin is quite clear, the triangle inequality amounts to saying that
(2.1')
d(x, y) ::; d(x, z) + d(z, y)
for all x, y, z E IC. We shall often use the notation d( a, b) to prepare the reader
for the extensions of analysis to functions of several variables, i.e. defined on
a subset of a real vector space of finite dimension such as lR P , or to much
more general spaces (Appendix to Chap. III).
Archimedes' axiom might appear obvious: one can, like John D. Rockefeller, amass a billion 1910 dollars, at five grams of gold to the dollar, by
saving one dollar a day for a sufficiently long time. But this does not follow
from (I) and (II): the professional mathematicians, in general more competent than the amateurs as in all the sports, have long since invented strange
''totally ordered nonarchimedean fields" which satisfy (I) and (II) but not
(III). One does not meet them in our usual mathematics.
One of the consequences of (III) is that if a real number x satisfies x ::; y
for all strictly positive y, then x ::; 0, for if not there would exist an integer
n such that nx > 1, whence x> y with y = lin> O. Other formulations:
(III') for all a, b E lR with a < b, there exists a u E Q such that a < u < b.
(III") for all a E lR and r > 0, there exists an x E Q such that dCa, x) < r.
Let us first prove (III"), which is obvious if one accepts the decimal expression for real numbers. If not, one proceeds as follows. The late lamented
Archimedes provides us an integer p > 0 such that lip < r; one can thus
restrict to the case where r = lip. The inequality to resolve can be written
lpa - pxl < 1, i.e.
b -1 < y < b + 1,
where b = pa and where y = px is neither more nor less rational than x. We
shall even show that, in this case, one can choose y in Z. If b > 0, there are
integers n such that b < n; the least of these then satisfies n - 1 < b < n,
whence b - 1 < n < b + 1 as desired. If b = 0, one takes y = 0. If b < 0, one
reduces to resolving b' -1 < y' < b' + Ion putting b' = -b > ° and y' = -yo
Whence (III").