1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences and Series 181

to the left of Lz + c:. If {xn} is not bounded below, then by definition

n->oo lim inf Xn = --lim Xn = -oo

n->oo

Note The limits superior and inferior of { xn} should not be confused with

the least upper bound (lub) and greatest lower bound (glb ), respectively,

of the set { xn} *. These four numbers may not coincide.

Example Consider the sequence defined by

1

xn=l+(-1r+(-1r-

n

Here we have x1 = -1, xz = 2.5, x3 = -^1 / 3 , x 4 = 2.25, and so on. In

general, for n = 2m - 1 we get the subsequence

1

Xzm-l = - --t 0

2m-1

and for n 2m we get

1

Xzm = 2 + - --t 2

2m

In this case

Li = n->oo lim sup Xn = 2,

On the other hand,

Ai = lub {xn} * = 2.5,

(see Fig. 4.1).

Lz = n->oo lim inf x n = 0

Az =glb{xn}* = -1

Theorem 4.2 Every real sequence {xn} has a limit superior and a limit
inferior ..

Proof If {xn} is not bounded above, then Li = +oo by definition. Suppose

that there exists a real constant M such that Xn < M for all n. Define a

Dedekind cut (A, B) of the rationals in the following way: If Xn > a for

infinitely many n, let a E A, and if there are not infinitely many terms of

the sequence such that Xn > b, let b E B. The cut (A, B) defines a real

number c. We wish to show that c = limn->oo sup Xn.

-2 0 x

Fig. 4.1