190
III - Convergence: Continuous variables
of a ball B with centre a and of radius R are those of the open ball with
the same centre and radius. Indeed, if a point z lies on the circumference
then every open ball with centre z meets both Band C - B; so it is neither
interior nor exterior to them, though it is adherent to both Band C - B.
This argument extends to any set X: to the splitting of points of lR (resp.
C) into the exterior and adherent points of X there corresponds a splitting
of the adherent points to X into the interior and boundary points of X (or
of the complement of X: these are clearly the same). If one chooses for X a
circumference d( a, x) = R, then all the points of X are boundary points and
all those z ¢ X are exterior; no point is interior to X, and X = X.
If all the points of a set are interior to it one says that the set is open; this
definition, here again, tallies with that of Chap. II when applied to intervals
or balls, and with that which we have given Ii propos analytic functions in
nO 6 and in nO 19: an open subset G of C must, for any a E G, contain an
open ball with centre a. One should pay attention to the fact that, contrary
to the concept of a closed set, that of an open set is not the same in lR as in
C: a "ball" in lR is an interval, which is not a ball in C. The interval JO, l[ is
open in lR but not in C.
By definition, a set X C lR (resp. q is closed if and only if every point of
the complement Y of X in lR (resp. q is exterior to X, i.e. interior to Y in
lR (resp. C); in other words, the complement in lR (resp. C) of a set closed in
lR (resp. q is open in lR (resp. C), and conversely.
These sets possess other important properties that are easy to establish.
First, the union of any family (Ui ) of open sets is an open set: for a point
a E U Ui belongs to some Ui which contains a ball with centre a necessarily
contained in the union. On the other hand, the intersection of a finite family
of open sets is an open set; for a point a E n Ui belongs to each Ui , a set
which contains an open ball Bi with centre a, so that n Ui ~ n B i ; now it
is clear that the intersection of a finite number of open balls with centre a is
again an open ball with centre a. Recall that the radius of an open ball is,
by definition, strictly positive. The argument fails for an infinite intersection:
that of the intervals J - lin, l/n[ reduces to {O} (Archimedes' axiom).
The corresponding properties of closed sets: every intersection of closed
sets (Pi) is a closed set; the union of a finite number of closed sets is a closed
set. In the first case, the complement of the intersection is the union of complements, open, of Pi, so is open. In the second case, the complement of the
union is the intersection of the complements, open.
After these set-theoretic gymnastic exercises let us return to the limit
values of a function. A first property is that if f(x) tends to a limit u ·when
x E X tends to a, then, for every sequence of points Xn EX, the relation
(1.5)
limxn = a implies limf(xn) = u.
III - Convergence: Continuous variables
of a ball B with centre a and of radius R are those of the open ball with
the same centre and radius. Indeed, if a point z lies on the circumference
then every open ball with centre z meets both Band C - B; so it is neither
interior nor exterior to them, though it is adherent to both Band C - B.
This argument extends to any set X: to the splitting of points of lR (resp.
C) into the exterior and adherent points of X there corresponds a splitting
of the adherent points to X into the interior and boundary points of X (or
of the complement of X: these are clearly the same). If one chooses for X a
circumference d( a, x) = R, then all the points of X are boundary points and
all those z ¢ X are exterior; no point is interior to X, and X = X.
If all the points of a set are interior to it one says that the set is open; this
definition, here again, tallies with that of Chap. II when applied to intervals
or balls, and with that which we have given Ii propos analytic functions in
nO 6 and in nO 19: an open subset G of C must, for any a E G, contain an
open ball with centre a. One should pay attention to the fact that, contrary
to the concept of a closed set, that of an open set is not the same in lR as in
C: a "ball" in lR is an interval, which is not a ball in C. The interval JO, l[ is
open in lR but not in C.
By definition, a set X C lR (resp. q is closed if and only if every point of
the complement Y of X in lR (resp. q is exterior to X, i.e. interior to Y in
lR (resp. C); in other words, the complement in lR (resp. C) of a set closed in
lR (resp. q is open in lR (resp. C), and conversely.
These sets possess other important properties that are easy to establish.
First, the union of any family (Ui ) of open sets is an open set: for a point
a E U Ui belongs to some Ui which contains a ball with centre a necessarily
contained in the union. On the other hand, the intersection of a finite family
of open sets is an open set; for a point a E n Ui belongs to each Ui , a set
which contains an open ball Bi with centre a, so that n Ui ~ n B i ; now it
is clear that the intersection of a finite number of open balls with centre a is
again an open ball with centre a. Recall that the radius of an open ball is,
by definition, strictly positive. The argument fails for an infinite intersection:
that of the intervals J - lin, l/n[ reduces to {O} (Archimedes' axiom).
The corresponding properties of closed sets: every intersection of closed
sets (Pi) is a closed set; the union of a finite number of closed sets is a closed
set. In the first case, the complement of the intersection is the union of complements, open, of Pi, so is open. In the second case, the complement of the
union is the intersection of the complements, open.
After these set-theoretic gymnastic exercises let us return to the limit
values of a function. A first property is that if f(x) tends to a limit u ·when
x E X tends to a, then, for every sequence of points Xn EX, the relation
(1.5)
limxn = a implies limf(xn) = u.