9.3 Implications of Uniform Convergence in Calculus 56500
Example 9.3.13 L f< sin~ converges uniformly on any compact interval to
k=l
00
a function whose derivative is .2:: ~ 2 cos ~.
k=lProof. Let a < b. We shall use Theorem 9.3.11 to prove that the given
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series converges uniformly as claimed on [a, b]. First, observe that L f< sin~
k=l
converges when x = 0. If 0 ~ [a, b], choose a' < b' so that 0 E [a', b'] and
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[a, b] ~ [a', b']. Next observe that the series of derivatives is L f2 cos~. By
k=l
the Weierstrass M-test, this series converges uniformly on [a', b']. Therefore, by
Theorem 9.3.11, our proof is complete. D
There is one more theorem of this type that we include here. It is a partial
converse of Corollary 9.3.6, which said that the uniform limit of a sequence of
continuous functions must be continuous. The theorem says that, under certain
circumstances, if the pointwise limit of a sequence of continuous functions is
continuous then the convergence must be uniform.
*Theorem 9.3.14 (Dini's Uniform Convergence Theorem) If Un} is a
sequence of continuous functions converging pointwise to a continuous function
f on a compact set S and if, \:/x E S , Un(x)} is monotone decreasing, then
f n -t f uniformly on S.
Proof. Suppose Un}, f, and S satisfy the hypotheses. For contradiction,
suppose the convergence fn -t f is not uniform on S. Then llfn -!II ft 0, so
:Jc> 0 3
II! n - f II > c, for infinitely many n. (7)
Since Sis compact, and since each Un(x)} is monotone decreasing, Inequality
(7) becomes
max{fn(x) - J(x) : x ES} > c, for infinitely many n.
Thus, there exists a sequence {xn} of points of S such that \:/n EN,
(8)By the Bolzano-Weierstrass theorem, {xn} has a convergent subsequence
{xnk}, say Xnk -t L. Since Sis compact, LES.
From Inequality (8), \:/k E N,
(9)