1550251515-Classical_Complex_Analysis__Gonzalez_

78 Chapter 2

in X, or as a sequence of elements of X, denoted {xn}, where Xn is the

value of the function at n. More specifically, we write {xn}~, or {xn};'°,
according to whether the sequence is finite or infinite. The value Xn is
called the nth term of the sequence. Of course, the terms x 1 , x 2 , x 3 ,
of a sequence need not be distinct.
The range of a sequence is the set of points

{xn}* = {x: x E X,x = Xn for some n}

The reader should not confuse a sequence with its range. For instance, if

{xn} = {O, 1, 0, 1, ... }, the range is the finite set {xn} * = {O, 1 }.

Let {xn};'° be an infinite sequence and {nk};_'° any strictly increasing

sequence of positive integers, i.e., such that ni < nz < ng < · ·" The

sequence {xnk}~ 1 is called a subsequence of {xn}:=l' This definition
implies that { x n} is a subsequence of itself. A similar definition can be
made for finite' sequences.
The definitions of union and of intersection of two sets can be extended
to any indexed collection of sets {Aa} as follows.

Definitions 2.4 The union of the sets Aa, a E I, is the set G such that
x E G · iff x E A°' for at least one a E J. In symbols,

G = LJ Aa = { x : x E Aa for some a E I}

a El

If I = JN, we usually write

N

G= LJ An

n=l

If I = J, we write

or

00

G= LJ An

n=l
The intersection of the sets A°' is the set H such that x E H iff x E A°'
for every a E J. In symbols,

H = n Aa = { x : x E A°' for every a E I}

a El

As for unions, we write

N

H = n An = Ai n Az n · · · n AN

n=l