1550251515-Classical_Complex_Analysis__Gonzalez_

182 Chapter4

To this end, choose arbitrarily a number a E A and a number b E B.
Then the interval (a, b] contains infinitely many terms Xn, since by the
definition of the classes A and B, there are infinitely many terms greater

than a but only a finite number greater than b. For any given E > 0 we

can choose a and b such that
c-e<a<c<b<c+e
and so the interval (c - e, c + e) contains infinitely many terms Xn. Since
there is only a finite number of terms greater than b, it ·follows that all
the terms satisfy Xn < c + e, except at most a finite number of them. In
other words, there exists N such that Xn < c + E for all n > N. Hence
c = limn-+oo sup Xn· The existence of the limit inferior is shown similarly.

Theorem 4.3 With the notations as in' Definitions 4.5 and 4.6, we have

L 2 :::; Li, and L 2 = Li iff {xn} has an ordinary limit L as n -+ oo, in

which case L = Li = Lz

Proof The assumption Li < L 2 violates the definitions of Li and L 2. In

fact, by choosing E <^1 / 2 (L2 - Li) we would have Li + E < Lz - E, so
that for n > N, say, all the terms would satisfy Xn < Li + E < L 2 - e,
contradicting the definition of Lz.

If Xn -+ L, then for a given E > 0 there is N such that n > N implies

that L - E <. Xn < L + E. But then L satisfies the definitions of both

Li and Lz, so that L = Li = L2. Conversely, if Li = L 2 , this number

is the ordinary limit of {xn}. In fact, given any E > 0, there is Ni such

that n > Ni implies that Xn < Li + E, and there is N 2 such that n > N 2

implies that Xn > Lz - E =Li - E. Hence for n > max(Ni,N 2 ) we have

Li - E < Xn <Li+ E, and it follows that limn-+oo Xn =Li = Lz.

4.5 Cauchy Condition for Convergence

Seldom can the convergence of a sequence of complex numbers be estab-
lished directly by using Definition 4.3, due to the difficulty involved in
determining its limit. Thus it is. convenient to have a rule or test that
will allow us to decide on the convergence of a sequence based only on the
knowledge of its terms. Such a rule is provided by the so-called Cauchy
condition (or criterion), an account of which has been given in Section 2.12.
As a particular case of Definition 2.41, we have

Definition 4. 7 A sequence {Zn} in (<C, d) is said to be a fundamental

sequence, or a Cauchy sequence, iff for every E > 0 there is N, such that

m, n > N, implies that

( 4.5-1)