566 Chapter 9 • Sequences and Series of Functions
Fix an no EN. 'Vx ES, Un(x)} is monotone decreasing, so whenever nk ~no,
f no ( Xnk) ~ f nk ( Xnk). Thus, from Equation ( 9) ,
nk ~no=> fn 0 (Xnk) - f(xnk) > c.
Taking the limit as k-. oo,
(10)
Since f n - f is continuous on S, Inequality (10) becomes
fn 0 (L) - J(L) ~ C > 0. (11)
We have shown that ( 11) is true for all no E N. Therefore, f n ( L) f; f ( L).
But {fn} converges pointwise to f on S, and LES. Contradiction. Therefore,
fn ___. f uniformly on S. •
EXERCISE SET 9.3
CXl
- The series I: xk converges uniformly on every closed interval [O, a] for
k=O
0 < a < 1 (Why?). Use Theorem 9.3.5 to prove that this series is not
uniformly convergent on [O, 1). - In each of the following, use Corollary 9.3.6 to prove that the given se-
quence does not converge uniformly on the given set.
(a) {1-xn} on[0,1] (b) {sinnx} on[O,n] (c)e-nx on[O,oo) - Find a sequence of functions that are discontinuous at every real number
that converges uniformly to a function that is not only continuous, but
differentiable at every real number.^6
nx - Let fn(x) = --. Show that even though {in} does not converge
l+nx
uniformly^7 on [O, 1], it is still true that lim f 0
1
fn = f 0
1
n--+oo n--+oo lim fn· Does
this violate Theorem 9.3.8? - 'tin E N, define fn on [O, 2] by f(x) = e-nx
2
- Find the pointwise limit
function f on [O, 2] and show that the convergence is not uniform. Nev-
ertheless, show that f~ f n -. f 02 f. [Hint: Theorem 9.3.8 applies to [c, 2]
when 0 < c < 2.]
- It is often said that uniform convergence is good at preserving good properties but bad at
preserving bad properties. - See Exercises 9.l.3(g) and 9.2.9(g).