550 Chapter 9 • Sequences and Series of Functions
Definition 9.2.3 (Uniform Convergence) We say that a sequence Un} of
functions in F(S, JR) converges uniformly to a function f E F(S, JR) if
I Ve> 0, :3 no EN 3 n 2 no=? Vx E S, lfn(x) - f(x)I < e. I
(Compare this with pointwise convergence in Exercise 3. See also Exercise 4.)Equivalently,
Ve> 0, :3 no EN 3 n 2 no=? fn - f is bounded on Sand llfn - !II < e.
That is,
I Un - !} is eventually in B(S) and from that point on,^4 llf n - f II -? 0. 1
We often indicate this by writing f n -? f (uniformly).
For bounded functions, the definition of uniform convergence can be stated
more simply.
Theorem 9.2.4 For a given sequence Un} of functions in B(S) and a function
f E B(S), {fn} converges uniformly to f if and only if llfn - !II -? 0.
That is,
..-I V_e_>-0,-:3-no_E_N_3_n_2:_n_o_=?_l_lf_n ___ f_l_I <-e-,.1Notice how closely this parallels the definition of lim Xn = x.
n->oo
Theorem 9.2.5 If fn(x)-? f (uniformly) on S, then fn(x)-? f (pointwise)
onS.
Proof. Exercise 6. •Examples 9.2.6 (a) As seen in Example 9.1.7 (a), the sequence {fn} of func-
tions fn(x) = xn converges pointwise on [O, 1] to the limit function
f(x ) = { 0 ~f 0:::; x < 1;
l1f x=l.
However, the convergence is not uniform, since
Vn EN, llfn - fll = sup{lxnl: 0 :=:; X :=:; 1} = 1, so llfn - fll f+ 0.
(b) Consider the same sequence Un} of functions as well as the same f
defined in (a), but consider convergence over the interval [O, c] where 0 < c < 1.
This time, llfn - fll = sup{lxnl : 0 :=:; x :=:; c} = en -? 0. Thus, fn(x) -? f
uniformly on [O, c].
- That is, ignoring the (finite number of) terms before which fn - f is bounded on S.