Suppose that there exists , such that for all (this will be true, in particular, if is cauchy). Prove that .
it seems like an easy proof, but i can't find the connection between limsup and cauchy sequence, need some help.
I will leave the epsilons to you but here is an idea. First there exists a subsequence
that converges to the limit supreemum and another subsequence
that converges to the lim inf
Now
Now just use the triangle inequality again and find the n,j, and k's to show what you want
I think that you have the wrong idea here. The given does not prove that the sequence is Cauchy. Rather it asks you to note that the conclusion would apply to Cauchy sequences.
First, you must prove the the both exist.
To do that notice that from the given
Thus they exist and if then .
Now you finish