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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?

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

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  • $\begingroup$ I have to know what is my problem,thanks your answer. Maybe if the sequence is monotone, the conclusion is right? And can we prove it? $\endgroup$ – David Chan Nov 26 '14 at 4:21
  • $\begingroup$ there may be different ways to make a correct version of your question, I do now know what you might possibly mean. If the sequence is monotone, then lim sup = lim, and indeed each subsequence has the same lim. But another possible (correct) variation is the following: The limit of a sequence (of real numbers) exists iff the limit superior of each subsequence exist, and lim sup of all subsequences are the same. You may dispose of all occurrences of sup and just say lim, in the previous sentence. But why should I speculate of what your question is supposed to ask? $\endgroup$ – Mirko Nov 26 '14 at 4:35

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