1550251515-Classical_Complex_Analysis__Gonzalez_

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4.4 Limit of a Real Sequence. Limit Superior and Limit Inferior

Chapter 4

In the particular case of a sequence { xn} of real terms the notation lim Xn =

L means: For every e > 0 there is an N (depending on e) such that n > N

implies that lxn - LI < e, or

L-€ < Xn < L + € (4.4-1)

so that all the terms of the sequence from a certain index on lie in the

open interval (L - e, L + e). Equivalently, if Xn _, L, all the terms of

the sequence, except for a finite number of them, are contained in a linear

€-neighborhood of L.

For real sequences this concept of limit (called the ordinary limit) can

be generalized by splitting into two the double inequality (4.4-1), and by

relaxing the condition that Xn is to satisfy on one of the two resulting

inequalities.

Definition 4.5 If {xn} is bounded above, i.e., if Xn < M (a constant)

for all n, we say that a real number Li is the limit superior of { Xn} as

n _, oo, and write

lim sup Xn = Li or

n-+oo

iff for every € > 0 there is N (depending on e) such that

for all n > N, while

for infinitely many values of n

Thus in this case all the terms of {xn}, except for a finite number of them,

lie to the left of Li + e, whatever be the given E > 0, while infinitely many

lie to the right of Li - e. If { Xn} is not bounded above, we write

lim sup Xn = lim Xn = +oo

n-+oo n-+oo

Definition 4.6 If { Xn} is bounded below, i.e., if Xn > M for all n, we say

that the real number L2 is the limit inferior of {xn} as n _, oo, and write

liminf n-+oo Xn = --lim Xn = L2

n-+oo

iff for every e > 0 there is N (depending on e) such that

for all n > N, while

for infinitely many values of n

Hence in the case of the limit inferior all the terms of { x n}, except for a
finite number of them, lie to the right of L 2 - e, while infinitely many lie