Limit Superior - Equivalent Definitions

In summary, the conversation discusses two different definitions of the limit superior, one from Rosenlicht's "Introduction to Analysis" and one as an intermediate definition. The conversation also explores the equivalence between the two definitions and suggests a connection between them.
  • #1
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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?
 
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  • #2
Try to work with the following intermediate definition:

[tex]\limsup a_n = \inf_n \sup_{m \ge n} a_m.[/tex]
 
  • #3
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.
 

1. What is the definition of limit superior?

The limit superior of a sequence is the largest value that the sequence approaches as n approaches infinity.

2. How is limit superior denoted?

The limit superior is denoted as lim sup or $\overline{\lim}$.

3. What is the equivalent definition of limit superior in terms of subsequences?

The limit superior of a sequence can also be defined as the largest limit of all possible subsequences of the original sequence.

4. Can the limit superior be infinite?

Yes, the limit superior can be infinite if the sequence does not have a largest limit as n approaches infinity.

5. Is the limit superior always defined for a sequence?

No, the limit superior is only defined for bounded sequences. If a sequence is unbounded, the limit superior is undefined.

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