How to Prove a Bounded Sequence {An} Converges to L if lim inf Equals lim sup?

[peripatein]

Hi,

What is the proof that if series {An} is bound and its lim inf = lim sup = L, then lim A must be equal to L?

 

[HallsofIvy]

What definition of "lim inf" and "lim sup" are you using?

 

[peripatein]

Lim inf = lowest partial limit, i.e. lowest limit amongst all the limits of all the sub-series.
Lim sup = highest partial limit, i.e. highest partial limit amongst all the limits of all sub-series

 

[HallsofIvy]

So you are saying that if a is a limit of any subsequence, then $A\le a\le A$! What does that tell you?

 

[peripatein]

a is not the limit of ANY subsequence, but the smallest of the limits of the subsequences and the largest of the limits of the subsequences.
How may I prove that the sequence itself converges to the same limit?