Limit Superior - Equivalent Definitions

[e(ho0n3]

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?

 

[morphism]

Try to work with the following intermediate definition:

$$\limsup a_n = \inf_n \sup_{m \ge n} a_m.$$

 

[e(ho0n3]

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.