Limit Superior - Equivalent Definitions

  • Thread starter e(ho0n3
  • Start date
  • #1
1,357
0
Let a_1, a_2, ... be a bounded sequence of real numbers. According to Rosenlicht's "Introduction to Analysis", the limit superior is defined as

sup {x : a_n > x for infinitely many n}.

It is very hard to work with this definition. I'm used to the simpler one:

sup {a : there exists a subsequence of a_1, a_2, ... that converges to a}.

I'm trying to show that these two are equivalent. Denote by A and B the set in the first and second definition, respectively. For any subsequence that converges to a in B, there is a monotonic subsequence, say b_1, b_2, ... that converges to a. If b_1, b_2, ... is increasing, then every b_i is in A. If it is decreasing, then x = inf {b_1, b_2, ...} is in A. For every x in A, there is a corresponding convergent monotonic subsequence. This is all I can think of. Any tips?
 

Answers and Replies

  • #2
morphism
Science Advisor
Homework Helper
2,015
4
Try to work with the following intermediate definition:

[tex]\limsup a_n = \inf_n \sup_{m \ge n} a_m.[/tex]
 
  • #3
1,357
0
That definition is just as abstruse as Rosenlicht's, at least to me. I thought about it a bit, but I don't see any connection with the other definitions. Here's another thought I have:

For any collection A' of terms of a_1, a_2, etc., inf A' is in A. So it seems to me that

sup A = sup { inf A' : A' is a collection of terms of a_1, a_2, ... }

This seems backwards from the definition you mentioned.
 

Related Threads on Limit Superior - Equivalent Definitions

Replies
5
Views
3K
  • Last Post
Replies
1
Views
6K
  • Last Post
Replies
3
Views
7K
  • Last Post
Replies
4
Views
3K
Replies
3
Views
4K
Replies
5
Views
4K
Replies
8
Views
10K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
7
Views
1K
Top