# 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?

• Have you given any thought to what "all subsequences" actually means? – Erick Wong Nov 26 '14 at 3:08
• @user48481MirkoSwirko Possibly the OP takes the convention that $\limsup$ exists if it belongs to $\mathbb R$? – Erick Wong Nov 26 '14 at 3:10
• @Erick Wong: all non-trival subsequences. – David Chan Nov 26 '14 at 3:11
• @DavidChan What counts as non-trivial? What about dropping the first term of the sequence? – Erick Wong Nov 26 '14 at 3:11
• @Erick Wong: Trivial means that if we exclude finite terms or a sequence, The remaining of the sequence is called a trivial subsequence. – David Chan Nov 26 '14 at 3:14