Convergence of a subsequence .

Question

If every subsequence of $x_n$ has a further subsequence which converges , is it true that the sequence is convergent?

NOTE : This is not a duplicate ofthis . In this problem it is not given that the subsequences converges to the same limit $x$.

Just take any bounded sequence which does not converge. – Aryabhata Jul 18 '14 at 05:50 — Aryabhata, Jul 18 '14 at 05:50

score 4 · Answer 1 · answered Jul 18 '14 at 04:21

4

Take the sequence $1,2,1,2,\ldots$ for a counterexample. It is not convergent, yet every subsequence either has an infinite number of ones or infinite number of twos and so has a convergent subsequence.

answered Jul 18 '14 at 04:21

please delete me

411

OK thanks . Trivial . :P – Soham Jul 18 '14 at 04:22
What subsequences are supposed to be converging to here? – hexaflexagonal Jul 18 '14 at 04:22
Oh, just saw your edit. You're right. I was incorrectly thinking of a subsequence as needing to comprise consecutive terms. – hexaflexagonal Jul 18 '14 at 04:23

Convergence of a subsequence .

1 Answers1