1549901369-Elements_of_Real_Analysis__Denlinger_

552 Chapter 9 • Sequences and Series of Functions

using the squeeze theorem. In summary,

3 no E .N 3 m, n 2 no=> \fx ES, lfn(x) - f(x)I < c.

That is, fn -t f uniformly. •

Theorem 9.2.8 (Uniform Convergence Preserves Boundedness) If {fn}

is a sequence of functions in B(S) converging uniformly on S to a real-valued

function f, then f E B(S).

Proof. Suppose {f n} is a sequence of functions in B(S) converging uni-
formly on Stoa real-valued function f. By Definition 9.2.3, 3 no EM 3 \fx E S,
n 2 no=> lfn(x)-f(x)I < 1. Since fno is a bounded function, 3M> 0 3 \fx E
S , lfn 0 (x)I < M. So, \fx ES,

lf(x)I :=:; lf(x) - fn 0 (x)I + lfn 0 (x)I < 1 + M.

That is, f is bounded on S. •

Definition 9.2.9 A sequence (or family) of functions is said to be uniformly

bounded on a set S if there exists a positive real number M such that for all

functions fin the sequence (or family), llfll :::; M; that is , \fx ES, lf(x)I:::; M.

Theorem 9.2.10 Every uniformly convergent sequence of functions in B(S)

is uniformly bounded.

Proof. Suppose Un} is a sequence in B(S) and fn -t f uniformly. By

Theorem 9.2.8, f E B(S). So, 3 M > 0 3 \fx E S , lf(x)I :::; M. Since fn -t f

uniformly, 3 no E .N 3 n 2 no=> \fx ES, lfn(x) - f(x)I:::; 1. Then \fx ES,

n 2 no=> lfn(x)I:::; lfn(x) - f(x)I + lf(x)I:::; 1 + M.

Since each function Ji, h , · · · , fno is bounded on S , 3 M1, M2, · · · , Mn 0 > 0 3
llfill :=:; M1, llhll :=:; M2, · · · , llfnoll :=:; Mno· Then,
\fn E .N, \fx ES, lfn(x)I:::; 1 + max{M, M1, M2, · · · , Mn 0 }.
That is, Un} is uniformly bounded. •

UNIFORM CONVERGENCE OF SERIES
00
We say that a series I: fk of functions fk E F(S, JR) converges uni-
k=O
formly to a function f E F(S, JR) if its sequence of partial sums converges
uniformly to f. As in Chapter 8, the Cauchy criterion applies to series as well
as sequences.