232
III - Convergence: Continuous variables
Theorem 13". Let X be a set and (fn) a sequence of scalar functions defined on X. For it to converge uniformly on X, it is necessary and sufficient
that for all r > 0 there exists an integer N such that
(10.5)
The condition is obviously necessary: by definition, uniform convergence
to f means that dx(fn, f) tends to 0, and since the triangle inequality applies
as much to the distance dx as to the usual distance in C, the inequality (5)
is obtained exactly as in the classical case of a sequence of numbers.
Sufficiency: since for any x
Theorem 13 shows that limfn(x) = f(x) exists for all x E X, and, further,
that If(x) - fn(x)1 ::; r for n > N. Since N does not depend on x one
concludes that dx(f, fn) ::; r for all n > N, whence uniform convergence (no
hypothesis at all on X).
The same idea is at the base of Theorems 13, 13' and 13": for f(x) to
converge when x tends to a limit a, finite or not, it is necessary and sufficient
that, for all r > 0, the function f(x) should be constant to within r on a
neighbourhood of a.
The second proof of Theorem 13 rests on axiom (IV) of Chap. II, but
the theorem is in fact equivalent to it, Le. allows us to give a proof of it,
or, equivalently, as we have already seen, to prove Theorem 2 of Chap. II,
nO 9 directly, for increasing sequences. For let (un) be an increasing sequence
bounded above. There is an index p such that up + 1 majorises the sequence,
otherwise there would exist a uq > up + 1, then an U r > uq + 1 > up + 2, etc.,
and the sequence would not be bounded. For the same reason, there exists
an index q > p such that uq + 1/10 majorises the sequence, then an index
r > q such that ur +l/100 majorises the sequence, etc. For i,j > r, Ui and Uj
lie between U r and U r + 1/100, whence lUi - ujl < 1/100, which is Cauchy's
criterion for 1/100. Etc.The given increasing sequence then converges, thanks
to Theorem 13.
One could also use the BW Theorem directly, since it assures the existence of a convergent subsequence; it is clear that the full sequence, being
increasing, must converge to the same limit.
Theorem 13 also allows one to show immediately that all nonterminating decimal expansions actually correspond to a real number: starting from
rank p, all the terminating expansions obtained by truncating it are equal
to within lO-P , and so the sequence formed by these terminating expansions
satisfies Cauchy's criterion trivially.
Conversely, in the case of an arbitrary sequence, when, for all p, the Un
of sufficiently large index are equal to each other to within lO-P; if decimal
III - Convergence: Continuous variables
Theorem 13". Let X be a set and (fn) a sequence of scalar functions defined on X. For it to converge uniformly on X, it is necessary and sufficient
that for all r > 0 there exists an integer N such that
(10.5)
The condition is obviously necessary: by definition, uniform convergence
to f means that dx(fn, f) tends to 0, and since the triangle inequality applies
as much to the distance dx as to the usual distance in C, the inequality (5)
is obtained exactly as in the classical case of a sequence of numbers.
Sufficiency: since for any x
Theorem 13 shows that limfn(x) = f(x) exists for all x E X, and, further,
that If(x) - fn(x)1 ::; r for n > N. Since N does not depend on x one
concludes that dx(f, fn) ::; r for all n > N, whence uniform convergence (no
hypothesis at all on X).
The same idea is at the base of Theorems 13, 13' and 13": for f(x) to
converge when x tends to a limit a, finite or not, it is necessary and sufficient
that, for all r > 0, the function f(x) should be constant to within r on a
neighbourhood of a.
The second proof of Theorem 13 rests on axiom (IV) of Chap. II, but
the theorem is in fact equivalent to it, Le. allows us to give a proof of it,
or, equivalently, as we have already seen, to prove Theorem 2 of Chap. II,
nO 9 directly, for increasing sequences. For let (un) be an increasing sequence
bounded above. There is an index p such that up + 1 majorises the sequence,
otherwise there would exist a uq > up + 1, then an U r > uq + 1 > up + 2, etc.,
and the sequence would not be bounded. For the same reason, there exists
an index q > p such that uq + 1/10 majorises the sequence, then an index
r > q such that ur +l/100 majorises the sequence, etc. For i,j > r, Ui and Uj
lie between U r and U r + 1/100, whence lUi - ujl < 1/100, which is Cauchy's
criterion for 1/100. Etc.The given increasing sequence then converges, thanks
to Theorem 13.
One could also use the BW Theorem directly, since it assures the existence of a convergent subsequence; it is clear that the full sequence, being
increasing, must converge to the same limit.
Theorem 13 also allows one to show immediately that all nonterminating decimal expansions actually correspond to a real number: starting from
rank p, all the terminating expansions obtained by truncating it are equal
to within lO-P , and so the sequence formed by these terminating expansions
satisfies Cauchy's criterion trivially.
Conversely, in the case of an arbitrary sequence, when, for all p, the Un
of sufficiently large index are equal to each other to within lO-P; if decimal