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Suppose we consider a sequence whose all subsequences converges to the same limit then can it be claimed that the original sequence also converges to the same limit? And a mathematical proof will be helpful for the argument.

Dheeraj Tiwari
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    Notice that the original sequence is also a subsequence. Therefore, if "all" the subsequences converge to the same limit, then the original sequence also converges to the limit! – Aniruddha Deshmukh Sep 11 '19 at 14:54
  • I suspect the statement is not what was intended. See my answer below. – copper.hat Sep 11 '19 at 15:04

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The result is true is a little more generality.

The result is a rather technical one, but is useful for some proofs.

Then $x_n \to L$ iff any subsequence contains a further subsequence that converges to $L$.

One direction follow from the definition.

Now suppose the other condition holds. If $x_n$ does not converge to $L$ then there is some $\epsilon>0$ and a subsequence $x_{n_k}$ such that $|x_{n_k} -L| > \epsilon$ for all $k$. It should be clear that this subsequence cannot contain a further subsequence converging to $L$, hence a contradiction.

copper.hat
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