1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences and Series

3. limz->zo,zEB Un(z) = Ln.

Then

(a) limn->oo Ln = L (L finite)

(b) limz->zo,zEB f(z) = L

199

The first row and first column in the following diagram show the hypoth-
esis in the theorem, while the second row and second column correspond
to the conclusion:

n->oo

un(z) =====f f(z)

z->zo zEB 1 1 z->zo zEB

Ln ------t L

n->oo

Proof (a) Since {un(z)} converges uniformly on B to f(z), by the Cauchy

condition we have, for every E > 0,

lum(z) - un(z)I < f for all z EB (4.10-1)

provided that m, n > N,. Hence letting z --+ zo (z remaining in B)
in (4.10-1), we obtain

ILm - Lnl ::; f

form, n > N,. Thus the sequence {Ln} satisfies the Cauchy condition for

every E > 0, so {Ln} converges to a finite limit L.

(b) Now, from the identity

J(z) - L = f(z) - un(z) + un(z) - Ln + Ln - L

it follows that

IJ(z) - LI ::; lf(z) - un(z)I + lun(z) - Lnl + ILn - LI

Given E > 0 we have, by assumption (1),

f

Jf(z) - Un(z)I < 3 for n > N 1 and z E B.

and by assumption (3),

forJz-zol<o(E), zEB

Also, we have, by the first part of the conclusion,

( 4.10-2)

( 4.10-3)

( 4.10-4)

(4.10-5)