§l. The intermediate value theorem
191
For any r > 0 there is an r' > 0 such that, for x EX, the inequality
d(x, a) < r' implies d[f(x), u] < r; but for n large one has d(xn' a) < r' and
so d[f(xn ), u] < r, qed.
Conversely one can show that if, for every sequence Xn E X converging
to a, the sequence of values J(xn ) converges to a limit which, a priori, depends on the sequence considered, then that limit is in reality independent
of it, and J(x) tends to the latter when x E X tends to a. This converse
is rarely used, but since it may oblige the reader to revise his ideas of
Socratic logic, always useful, let us give the proof.
The fact that the limit is always the same is immediate: if J(xn ) tends
to u, if J(Yn) tends to v and if one denotes by (Zn) the sequence
obtained by interlacing the two given sequences, it tends to a as do the
two chosen sequences, so the sequence J(Zn) can converge only if u = v. It
remains to show that J(x) tends to this limit u common to all the sequences
considered. We argue by contradiction, assuming this assertion false. Now
the relation limJ(x) = u means that "for all r > 0 there exists an r' > 0"
having a certain property. If such is not the case, there exists an r > 0 such
that, Jor all r' > 0, the property in question is false. This property is that
the relations x E X and Ix - al < r' imply IJ(x) - ul < r. The negation
of this property is that there exists an x E X satisfying Ix - al < r' while
not satisfying IJ(x) - ul < r, so satisfying IJ(x) - ul ~ r.
Applying this argument to r' = 1, 1/2, 1/3, etc., one finds a sequence
of points Xn E X satisfying
IXn - al < lin and IJ(xn ) - ul ~ r for all n.
In other words, contrary to the hypothesis, there exists a sequence Xn E X
which converges to a but for which J(xn ) does not converge to u, qed.
A second type of property of limits COncerns what happens when one performs simple algebraic operations on functions which tend to limits. We have
to distinguish several cases; we restrict to the case of a real variable which
increases indefinitely, the statements and proofs being practically identical
in the others, both in lR and in C. The theory is really nothing new, since
the essential ideas, if one can speak of ideas, have already been explained for
sequences.
Theorem 1. Let f and 9 be scalar functions defined on a set X c lR which is
not bounded above, and suppose that f(x) and g(x) tend to u and v as x E X
increases indefinitely. Then f(x) + g(x) and f(x)g(x) tend respectively to
u + v and uv. If v 1= 0 then g(x) 1= 0 for x large and the function f(x)/g(x)
tends to u/v.
The proofs are those of Theorem 1 of Chap. II, nO 8: one replaces Un and
Vn by f(x) and g(x) and "for n large" by "for x large", or, in the case where
x tends to a finite a, by "for x sufficiently close to a" . In particular, and this
191
For any r > 0 there is an r' > 0 such that, for x EX, the inequality
d(x, a) < r' implies d[f(x), u] < r; but for n large one has d(xn' a) < r' and
so d[f(xn ), u] < r, qed.
Conversely one can show that if, for every sequence Xn E X converging
to a, the sequence of values J(xn ) converges to a limit which, a priori, depends on the sequence considered, then that limit is in reality independent
of it, and J(x) tends to the latter when x E X tends to a. This converse
is rarely used, but since it may oblige the reader to revise his ideas of
Socratic logic, always useful, let us give the proof.
The fact that the limit is always the same is immediate: if J(xn ) tends
to u, if J(Yn) tends to v and if one denotes by (Zn) the sequence
obtained by interlacing the two given sequences, it tends to a as do the
two chosen sequences, so the sequence J(Zn) can converge only if u = v. It
remains to show that J(x) tends to this limit u common to all the sequences
considered. We argue by contradiction, assuming this assertion false. Now
the relation limJ(x) = u means that "for all r > 0 there exists an r' > 0"
having a certain property. If such is not the case, there exists an r > 0 such
that, Jor all r' > 0, the property in question is false. This property is that
the relations x E X and Ix - al < r' imply IJ(x) - ul < r. The negation
of this property is that there exists an x E X satisfying Ix - al < r' while
not satisfying IJ(x) - ul < r, so satisfying IJ(x) - ul ~ r.
Applying this argument to r' = 1, 1/2, 1/3, etc., one finds a sequence
of points Xn E X satisfying
IXn - al < lin and IJ(xn ) - ul ~ r for all n.
In other words, contrary to the hypothesis, there exists a sequence Xn E X
which converges to a but for which J(xn ) does not converge to u, qed.
A second type of property of limits COncerns what happens when one performs simple algebraic operations on functions which tend to limits. We have
to distinguish several cases; we restrict to the case of a real variable which
increases indefinitely, the statements and proofs being practically identical
in the others, both in lR and in C. The theory is really nothing new, since
the essential ideas, if one can speak of ideas, have already been explained for
sequences.
Theorem 1. Let f and 9 be scalar functions defined on a set X c lR which is
not bounded above, and suppose that f(x) and g(x) tend to u and v as x E X
increases indefinitely. Then f(x) + g(x) and f(x)g(x) tend respectively to
u + v and uv. If v 1= 0 then g(x) 1= 0 for x large and the function f(x)/g(x)
tends to u/v.
The proofs are those of Theorem 1 of Chap. II, nO 8: one replaces Un and
Vn by f(x) and g(x) and "for n large" by "for x large", or, in the case where
x tends to a finite a, by "for x sufficiently close to a" . In particular, and this