the limit superior of a sequence exists iff the limit inferior of all subsequences of the sequence exist? 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?

Please clarify your question, as stated the answer is obviously no (and you should know).

" _... 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._ "

The above is wrong! Take the sequence 0,1,0,1.... Then lim sup exists and equals 1. The lim sup of the subsequence 0,0,0... is 0, and the lim sup of the subsequence 1,1,1... is 1, and these two are not equal to each other. So, even though the lim sup of the given sequence exists, it is not true that each subsequence has the same lim sup. Clearly this contradicts your iff statement.

Perhaps you need to change some lim sup and some lim inf in your question with simply lim, in order for it to make sense?