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Actually, I found the same questions in this website and saw the proof.

What I'm looking for is somewhat intuitive understanding.

I know that, {$x_n$} converges to {$x$} if and only if every subsequence {$x_{n_k}$} converges to {$x$}.

So, if {$x_n$} converges to {$x$}, it looks trivial that every subsequence has a sub-subsequence that converges to {$x$}. Actually all the sub-subsequence converges to the same point, I think.

However, the opposite way is not easy for me. How can the existence of a convergent sub-subsequence in each subsequence guarantees the convergence of sequence?

For me, it looks like that in each subsequence, the existence of one sub-subsequence which converges guarantees the convergence of all sub-subsequence.

What am I missing?

smw1991
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  • Check this link http://math.stackexchange.com/questions/397978/every-subsequence-of-x-n-has-a-further-subsequence-which-converges-to-x-then – Truong Mar 31 '16 at 04:02

3 Answers3

3

Suppose your sequence does not converge to $x$. Therefore, there exists a neighbourhood of $x$ which, no matter how big $N$, there is always some $x_n$ for $n>N$ such that $x_n$ is not on that neighbourhood. This way, you can construct a subsequence which never enters that neighbourhood. But this subsequence will not be able to have any subsequence converging to $x$, by its construction.

Aloizio Macedo
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1

There is really no need for a proof of any type if you know that any convergent sequence shares its limit with any subsequence. Let $A$ be a convergent sequence and $B$ be any subsequence of $A$. Let $C$ be any subsequence of $B$. Then $B$ and $C$ share the same limit, therefore so does $C$ and $A$ and you are done. The main thing is that a subsequence of a subsequence of a sequence is a subsequence of the original sequence. That is a bit of a mouthful, but try to understand that statement.

Alex Ortiz
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Faraad Armwood
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0

It is because the sequence itself is an actual subsequence of itself. So for all the subsequences to converge, the actual sequence has to converge to. (and similarly for a sub-subsequence).

Inazuma
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