1549901369-Elements_of_Real_Analysis__Denlinger_

9.3 Implications of Uniform Convergence in Calculus 561

Theorem 9.3.8 (Uniform Convergence Preserves Integrals) If fn--> f

uniformly on a closed interval [a, b], and if each f n is integrable on [a, b], then

f is integrable on [a, b] and t a f = n--+oo lim t a fn·

Proof. Suppose fn --> f uniformly on [a, b], and each fn is integrable on

[a, b]. Note that 't:/x E [a, b], lfn(x) - f(x)I ~ llfn -!II , so

fn(x) - llfn - fll ::; f(x)::; fn(x) + llfn -!II.

Then l: Un - II! n - !II) = l: Un - II! n - f II) ~ l: f ::; l: f

~ l: Un+ llfn -!II)= l: Un+ llfn - !II). (3)

So, 0 ~ l: f - l: f ::; l: Un+ II! n - f II) - l: Un - II! n - !II)

0 ~ l: f - l: f ~ 2 l: llfn -!II = 2 llfn - !II (b - a).

Since f n --> f uniformly on [a, b], II! n - !II --> 0. So, when we take the limit
as n --> oo, we have

which means that f is integrable on [a, b].
From inequality (3) we have

(I: fn) - llfn - fll (b - a)~ l: f ~ (f: fn) + llfn -Jll (b - a)

• II! n - !II (b - a) ::; l: f - l: f n ::; II! n - f II (b - a)

11 : f - l: fnl ~ llfn - fll (b - a)--> 0.

Therefore, by the second squeeze principle (2.3.2), n--+oo lim t a fn = t a f. •

Corollary 9.3.9 If a series 2: fk of integrable functions converges uniformly

to f on [a, b], then f is integrable on [a, b], and l: f = L l: fk.

That is, if 2: fk converges uniformly, then l: L fk = L l: fk.

1

rr sinnx
Example 9.3.10 Evaluate lim --dx, where 0 < a < 1r.
n->oo a nx