The Real And Complex Number Systems

Supplement on lim sup and lim inf
Introduction
In order to make us understand the information more on approaches of a given real
sequence a n
n1
, we give two definitions, thier names are upper limit and lower limit. It
is fundamental but important tools in analysis. We do NOT give them proofs. The reader
can see the book, Infinite Series by Chao Wen-Min, pp 84-103. (Chinese Version)
Definition
Definition of limit sup and limit inf
and
Example
Example
Example
Given a real sequence a n
n1
,wedefine
b n supa m : m ≥ n
1 −1 n
n1
−1 n
n n1
−n n1
c n infa m : m ≥ n.
0,2,0,2,..., sowehave
b n 2andc n 0 for all n.
−1, 2, −3,4,..., sowehave
b n and c n − for all n.
−1, −2, −3, . . . , sowehave
b n −n and c n − for all n.
Proposition Given a real sequence a n
n1
, and thus define b n and c n as the same as
before.
1 b n ≠−,andc n ≠∀n ∈ N.
2 If there is a positive integer p such that b p , then b n ∀n ∈ N.
If there is a positive integer q such that c q −, then c n − ∀n ∈ N.
3 b n is decreasing and c n is increasing.
By property 3, we can give definitions on the upper limit and the lower limit of a given
sequence as follows.
Definition Given a real sequence a n and let b n and c n as the same as before.
(1) If every b n ∈ R, then
infb n : n ∈ N
is called the upper limit of a n , denoted by
lim n→
sup a n .
That is,
lim n→
sup a n inf b n.
n
If every b n , then we define
lim n→
sup a n .
(2) If every c n ∈ R, then
supc n : n ∈ N
is called the lower limit of a n , denoted by