60
II - Convergence: Discrete variables
an inequality d( a, x) < r with a strictly positive r, in other words, in the case
of JR, any interval]a - r, a + r[ with r > 0 and, in the case of C, any disc of
centre a, the circumference excluded. Then (3.1) means that there exists an
open ball B(a, r) = B with centre a such that
x E BnE ===} P(x).
The fact that we have chosen "open" balls here, i.e. defined by a strict inequality d( a, x) < r, rather than closed balls defined by a weak inequality
d( a, x) :::; r, is of no importance: an open ball with centre a and radius r contains every closed ball with the same centre and radius r' < r, and vice-versa;
a relation valid in an open ball will also be valid in a smaller closed ball and
vice-versa.
Most authors, following a tradition that goes back, at least, to the famous
courses in analysis delivered by Karl Weierstrass in Berlin around 1870, use c
or fJ for what we call r, the psychology of this notation being that these letters
are reserved for ''very small" numbers; it is obvious that if one can verify that
the assertion P(x) is true once d(a,x) < 10- 4 it is unnecessary to examine
what happens in the ball d( a, x) < 1000. This usage, however, probably stems
from the fact that fJ is the initial letter of the word "difference" and that c
immediately follows fJ in the Greek alphabet; we realise this when we define
the continuity of a function at a:
for any c > 0 there exists a fJ > 0 such that
Ix - al < fJ ===} If(x) - f(a)1 < c,
or again: if the difference between x and a is sufficiently small, i.e. smaller
than a suitably chosen number fJ > 0, then the difference between f(x) and
f(a) is also as small as one wishes, i.e. smaller than any number c > 0
given in advance. The use of the letter fJ (apparently begun by Cauchy) is
thus relatively rational, and that of the letter c (introduced by Weierstrass)
probably followed for a reason quite unconnected with mathematics.
In any case, r (or c, or fJ) may also be "very large" since one asks no more
than they exist; further, the concept of a "very small" or ''very large" fixed
number has no objective meaning. There could be no objection if the reader
followed a preference for the letter R, or p, or whatever he wanted, a 0 or $
sign for example: in a statement such as (1), as in the notation L in nO 1,
the letter r is a phantom, a "bound variable" , whose only role is to serve as
the logical link between the assertions "there exists r > 0" and "Ix - al < r
implies P(x)". We prefer the letter r because it suggests the radius of a ball;
moreover, it is directly available on all typewriter and computer keyboards.
One might also restrict oneself to powers of 10 and say:
there exists an n E Z such that P(x) for all x E E satisfying
Ix - al < lO- n .
II - Convergence: Discrete variables
an inequality d( a, x) < r with a strictly positive r, in other words, in the case
of JR, any interval]a - r, a + r[ with r > 0 and, in the case of C, any disc of
centre a, the circumference excluded. Then (3.1) means that there exists an
open ball B(a, r) = B with centre a such that
x E BnE ===} P(x).
The fact that we have chosen "open" balls here, i.e. defined by a strict inequality d( a, x) < r, rather than closed balls defined by a weak inequality
d( a, x) :::; r, is of no importance: an open ball with centre a and radius r contains every closed ball with the same centre and radius r' < r, and vice-versa;
a relation valid in an open ball will also be valid in a smaller closed ball and
vice-versa.
Most authors, following a tradition that goes back, at least, to the famous
courses in analysis delivered by Karl Weierstrass in Berlin around 1870, use c
or fJ for what we call r, the psychology of this notation being that these letters
are reserved for ''very small" numbers; it is obvious that if one can verify that
the assertion P(x) is true once d(a,x) < 10- 4 it is unnecessary to examine
what happens in the ball d( a, x) < 1000. This usage, however, probably stems
from the fact that fJ is the initial letter of the word "difference" and that c
immediately follows fJ in the Greek alphabet; we realise this when we define
the continuity of a function at a:
for any c > 0 there exists a fJ > 0 such that
Ix - al < fJ ===} If(x) - f(a)1 < c,
or again: if the difference between x and a is sufficiently small, i.e. smaller
than a suitably chosen number fJ > 0, then the difference between f(x) and
f(a) is also as small as one wishes, i.e. smaller than any number c > 0
given in advance. The use of the letter fJ (apparently begun by Cauchy) is
thus relatively rational, and that of the letter c (introduced by Weierstrass)
probably followed for a reason quite unconnected with mathematics.
In any case, r (or c, or fJ) may also be "very large" since one asks no more
than they exist; further, the concept of a "very small" or ''very large" fixed
number has no objective meaning. There could be no objection if the reader
followed a preference for the letter R, or p, or whatever he wanted, a 0 or $
sign for example: in a statement such as (1), as in the notation L in nO 1,
the letter r is a phantom, a "bound variable" , whose only role is to serve as
the logical link between the assertions "there exists r > 0" and "Ix - al < r
implies P(x)". We prefer the letter r because it suggests the radius of a ball;
moreover, it is directly available on all typewriter and computer keyboards.
One might also restrict oneself to powers of 10 and say:
there exists an n E Z such that P(x) for all x E E satisfying
Ix - al < lO- n .