I have already proved that if ${X_k}$ converges to a limit $L$, then any subsequence of it also converges to $L$. And now the question asks to show that if ${X_k}$ has two subsequence which converge to two different limits, then ${X_k}$ can not be convergent.
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1Have you proved that a sequence can't converge to two different limits? – bof Oct 16 '13 at 22:00
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@bof: not yet. Should I prove it first? – Iceyyy Oct 16 '13 at 22:05
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I believe you are getting a little confused in the logic of the question. If you think it through, when you say that you have already proved that if a sequence converges, then every subsequence converges to the same limit, you can readily answer the question using just that.
To see this more carefully, argue by contradiction. Assume that the given sequence has two subsequences that converge to different limits AND suppose for the sake of contradiction that the original sequence is also convergent. What to do now? Well, since you are assuming that the sequence is convergent, by what you said you proved,then every subsequence converges to the same limit. In particular, the two subsequences must converge to the same limit aswell...but you had said before that they converged to different limits! Contradiction!!
Thus, the original sequence cannot be convergent.
Luke Skywalker
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Hint: Assume the sequnce converges to some limit $L$. What can be said about the so-called different limits of the subsequences.
Hagen von Eitzen
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But the condition is given that two subsequence converges to different limit – Iceyyy Oct 16 '13 at 22:28
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A sequence $\{ x_n \}$ converges to $L$ if and only if every subsequence of $\{ x_n \}$ converges to $L$. Therefore, if there exists two subsequences $\{ x_{n_k} \}$ and $\{ x_{n_l} \}$ converging to two different limits $L'$ and $L''$, then $\{ x_n \}$ cannot be convergent.
tylerc0816
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