110
II - Convergence: Discrete variables
the value 1 + 1/3 + 1/3.4 - 1/3.4.34, or 1.4142157 for the square root of 2,
equal to 1.4142136 ... By a coincidence as surprising as unexpected, one has
17.17 + 12.24 _ 2.17.17 -1 _ 7/ _ /3
_
24.17
-
2.3.4.17 - 1 3.4 1 .4.34(12 + 4 + 1)/3.4 - 1/3.4.34 = 1 + 1/3 + 1/3.4 - 1/3.4.34.
11 - What is an integral?
The concept of an integral is a particularly striking and fundamental application of axiom (IV); it is not difficult to introduce it here, without waiting
for Chap. V.
First we note that one can state axiom (IV) as follows:
(IV bis) Let E and F be two nonempty sets of real numbers. Suppose
that every u E E is smaller than every v E F. Then there exists a
number w such that u ::; w ::; v for all u E E and v E F. For w to
be unique it is necessary and sufficient that for every number r > 0
there exist a u E E and a v E F such that v - u < r.
It is clear that E is bounded above, so possesses a least upper bound m, and
that F is bounded below, so possesses a greatest lower bound M. Since every
v E F majorises E, one has m ::; v. In consequence, m minorises F, whence
m ::; M (which justifies the notation ... ). Every w E [m, M] will then do.
Finally, for w to be unique, it is necessary and sufficient that m = M; and
then one can always find an u E E and an v E F arbitrarily close to m and
M, whence the second point.
This done, let us return to integrals. The problem is the following: given a
function f(x) with real values (for simplicity) on a compact interval I = [a, b],
one wants to give a meaning to the measure
m(f) = lb f(x)dx
of the area between the x-axis, the graph of f and the verticals x = a and
x = b, it being understood that one counts as positive the areas lying above
the x-axis and the others as negative, so as to obtain the linearity relation
m(af + f3g) = am(f) + f3m(g) for any constants a and f3.
in their time. It has also been known for a long time that in Newton's time,
the Japanese, for their part, were in process of making very similar discoveries
to his. The reasons why all these advances were transformed into an enormous
delay behind the Westerners might form the subject of all sorts of theories and
are surely not the same in Japan, which isolated herself voluntarily in order
to escape possible colonisation, as in the Arab countries of the Middle East or
of Central Asia; see Bernard Lewis, The Muslim Discovery of Europe (Norton,
1982). In both cases, the interest in Western Civilisation first came from Western
weapons ...
II - Convergence: Discrete variables
the value 1 + 1/3 + 1/3.4 - 1/3.4.34, or 1.4142157 for the square root of 2,
equal to 1.4142136 ... By a coincidence as surprising as unexpected, one has
17.17 + 12.24 _ 2.17.17 -1 _ 7/ _ /3
_
24.17
-
2.3.4.17 - 1 3.4 1 .4.34(12 + 4 + 1)/3.4 - 1/3.4.34 = 1 + 1/3 + 1/3.4 - 1/3.4.34.
11 - What is an integral?
The concept of an integral is a particularly striking and fundamental application of axiom (IV); it is not difficult to introduce it here, without waiting
for Chap. V.
First we note that one can state axiom (IV) as follows:
(IV bis) Let E and F be two nonempty sets of real numbers. Suppose
that every u E E is smaller than every v E F. Then there exists a
number w such that u ::; w ::; v for all u E E and v E F. For w to
be unique it is necessary and sufficient that for every number r > 0
there exist a u E E and a v E F such that v - u < r.
It is clear that E is bounded above, so possesses a least upper bound m, and
that F is bounded below, so possesses a greatest lower bound M. Since every
v E F majorises E, one has m ::; v. In consequence, m minorises F, whence
m ::; M (which justifies the notation ... ). Every w E [m, M] will then do.
Finally, for w to be unique, it is necessary and sufficient that m = M; and
then one can always find an u E E and an v E F arbitrarily close to m and
M, whence the second point.
This done, let us return to integrals. The problem is the following: given a
function f(x) with real values (for simplicity) on a compact interval I = [a, b],
one wants to give a meaning to the measure
m(f) = lb f(x)dx
of the area between the x-axis, the graph of f and the verticals x = a and
x = b, it being understood that one counts as positive the areas lying above
the x-axis and the others as negative, so as to obtain the linearity relation
m(af + f3g) = am(f) + f3m(g) for any constants a and f3.
in their time. It has also been known for a long time that in Newton's time,
the Japanese, for their part, were in process of making very similar discoveries
to his. The reasons why all these advances were transformed into an enormous
delay behind the Westerners might form the subject of all sorts of theories and
are surely not the same in Japan, which isolated herself voluntarily in order
to escape possible colonisation, as in the Arab countries of the Middle East or
of Central Asia; see Bernard Lewis, The Muslim Discovery of Europe (Norton,
1982). In both cases, the interest in Western Civilisation first came from Western
weapons ...