228
III - Convergence: Continuous variables
f(a) ::; f(x) ::; f(b) for all x E K.
The image f(K) is certainly a compact subset of lR.. It is bounded, so
that u = inf(f(K)) and v = sup(f(K)) are finite. Since f(K) is closed it
must contain u and v. So there are points a, bE K for which f(a) = u and
f(b) = v, qed.
The preceding corollary, which states that f has an absolute minimum and
maximum on K, can be generalised to the case of continuous functions with
complex values: for the equation f(x) = w to possess an exact solution x E K
for w E C given, it is sufficient that it possesses a solution to within lO- n
for all n E N, i.e. that there exists an Xn E K such that If(xn ) - wi < lO- n .
For if so, then w is adherent to f(K), so belongs to this set, because f(K) is
compact and therefore closed.
Theorem 12. Let f be a complex-valued junction, defined, continuous and
injective on a compact set K c C. Then g : f(K) -+ K, the inverse map to
f, is continuous.
Let b = f(a) be a point of H = f(K), choose an r > 0 and consider the
set K' of x E K such that d( a, x) ~ r. This is closed and bounded like K,
so compact. In consequence, H' = f(K') is a compact subset of C (in fact of
H) and in particular is closed. Since f is injective, H' does not contain b, so
there exists an r' > 0 such that the ball B(b, r') does not meet H'. It is clear
that then
f(x) E B(b,r') ===} x E B(a,r),
qed.
10 - Cauchy's general convergence criterion
Theorem 13. (Cauchy's general convergence criterion for sequences)
For a sequence (un) of complex numbers to converge it is necessary and sufficient that for all r > 0
(10.1)
for p and q sufficiently large.
First proof We use the BW theorem. By this there exists a subsequence U(Pn) which converges to a limit u. For r > 0 given we thus have
IU(Pn) - ul < r for n large. But since Pn ~ n we also have, by hypothesis,
IU(Pn) - u(n)1 < r once n is sufficiently large. So lu(n) - ul < 2r for n large,
qed.
Second proof First we note that if the given sequence satisfies (1) then so
do the sequences formed by the real and imaginary parts of Un. So we can
restrict ourselves to the case of a sequence of real numbers.
III - Convergence: Continuous variables
f(a) ::; f(x) ::; f(b) for all x E K.
The image f(K) is certainly a compact subset of lR.. It is bounded, so
that u = inf(f(K)) and v = sup(f(K)) are finite. Since f(K) is closed it
must contain u and v. So there are points a, bE K for which f(a) = u and
f(b) = v, qed.
The preceding corollary, which states that f has an absolute minimum and
maximum on K, can be generalised to the case of continuous functions with
complex values: for the equation f(x) = w to possess an exact solution x E K
for w E C given, it is sufficient that it possesses a solution to within lO- n
for all n E N, i.e. that there exists an Xn E K such that If(xn ) - wi < lO- n .
For if so, then w is adherent to f(K), so belongs to this set, because f(K) is
compact and therefore closed.
Theorem 12. Let f be a complex-valued junction, defined, continuous and
injective on a compact set K c C. Then g : f(K) -+ K, the inverse map to
f, is continuous.
Let b = f(a) be a point of H = f(K), choose an r > 0 and consider the
set K' of x E K such that d( a, x) ~ r. This is closed and bounded like K,
so compact. In consequence, H' = f(K') is a compact subset of C (in fact of
H) and in particular is closed. Since f is injective, H' does not contain b, so
there exists an r' > 0 such that the ball B(b, r') does not meet H'. It is clear
that then
f(x) E B(b,r') ===} x E B(a,r),
qed.
10 - Cauchy's general convergence criterion
Theorem 13. (Cauchy's general convergence criterion for sequences)
For a sequence (un) of complex numbers to converge it is necessary and sufficient that for all r > 0
(10.1)
for p and q sufficiently large.
First proof We use the BW theorem. By this there exists a subsequence U(Pn) which converges to a limit u. For r > 0 given we thus have
IU(Pn) - ul < r for n large. But since Pn ~ n we also have, by hypothesis,
IU(Pn) - u(n)1 < r once n is sufficiently large. So lu(n) - ul < 2r for n large,
qed.
Second proof First we note that if the given sequence satisfies (1) then so
do the sequences formed by the real and imaginary parts of Un. So we can
restrict ourselves to the case of a sequence of real numbers.