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.
that converges to the limit supreemum and another subsequence
that converges to the lim inf
Now just use the triangle inequality again and find the n,j, and k's to show what you want
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