The question is nearly the same as the title, that is, the limit superior of a sequence (of real numbers) exists (can be infinity)iff the limit superior of all subsequences of the previously mentioned sequence exist, and converge to the same limit , and the limit inferior of a sequence exists iff the limit inferior of all sequence of the sequence exist converge to the same limit ?
I have known that if the limit superior of a sequence exist, then the limit superior of all subsequences of the previously mentioned sequence exist and converge to the same limit , too. But what about the reverse?
Another question, if the limit superior of a sequence exists and is a positive number, can we conclude that any subsequence of this sequence is positive when independent variable n is bigger enough?