Every subsequence of xnx_n has a further subsequence which converges to xx. Then the sequence xnx_n converges to xx.

Is the following true?
Let xn be a sequence with the following property: Every subsequence of xn has a
further subsequence which converges to x. Then the sequence xn converges to x.

I guess that it is true but I am not sure how to prove this.

Answer

True. If not, there exists an ϵ>0, such that for all k, there exists an nk>k satisfying |xnkx|ϵ since if there is some k which doesn’t have such nk, then we can take it as N, so xn converges to x. The subsequence xnk does not have any subsequence converging to x.

Attribution
Source : Link , Question Author : gulim , Answer Author : user124697

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