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Let $(x_n)$ be a sequence of real numbers.

Prove that if there exists $x$ such that every subsequence $(x_{n_k})$ of $(x_n)$ has a convergent (sub-)subsequence $(x_{n_{k_l}})$ to $x$, then the original sequence $(x_n)$ itself converges to $x$ .

Thanks for any help.

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What have you tried? Hint: try to show it by contradiction. – Davide Giraudo Jul 21 '12 at 12:23
2  
As you wrote it it can be a little misleading, imo. The claim is actually: a sequence converges to some limit $\,l\,$ iff every infinite subsequence converges to the very same limit $\,l\,$ . Going from here to subsequences of subsequences is easy, though a little messy with the sub-sub-indexes. – DonAntonio Jul 21 '12 at 12:27
    
@Davide I was trying to proceed by contradiction by making 2 cases- a(n) is bounded and unbounded . – Ester Jul 21 '12 at 12:29
    
Notice that you can have sequences where every subsequence has a convergent subsequence, but said subsequences have different limits, and (hence) the overall sequence does not converge... – Ben Millwood Jul 21 '12 at 15:26
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@DonAntonio Done, I have balanced the downvote. – 1015 Apr 9 '13 at 14:07
up vote 10 down vote accepted

First notice that your condition implies that your sequence is bounded.

Indeed, if $(x_n)$ is unbounded, we can find a subsequence $(x_{n_k})$ such that $|x_{n_k}|\ge k$. This subsequence does not have a convergent subsequence.


So we know that $(x_n)$ is bounded and it is not convergent. This means that $$M=\limsup x_n > \liminf x_n =m.$$ (Both $M$ and $m$ are real numbers, since $(x_n)$ is bounded.)

We know (from the properties of limit superior and limit inferior) that there is a subsequence $(x_{n_k})$ which converges to $M$ and there is a subsequence $x_{n_l}$ which converges to $m$. (And every subsequence of any of these two subsequences has, of course, the same limit $M$ resp. $m$.)

We have found two subsequences with different limits, which contradicts your assumptions about the sequence $(x_n)$.

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Thanks for your help. – Ester Jul 21 '12 at 12:33

Suppose $x_n$ does not converge to $x$, but every subsequence of $x_n$ has a sub-subsequence which converge to $x$.

Then, first we can see that $x_n$ must be bounded, because otherwise, we can easily construct a subsequence $x_{n_k}$ such that $|x_{n_k}|\rightarrow\infty$.

Since $x_n$ does not converge to $x$ we must be able to find a subsequence such that every term is more than $\epsilon$ away from $x$ for some $\epsilon>0$, but clearly this does not have a sub-subsequence which converges to $x$, by definition.

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The second paragraph appears to be extraneous. – Live Forever Nov 12 '15 at 1:09

Let every subsequence of x_n has a convergent subsequence to x and suppose by way of contradiction that x_n does not converges to x . Then there exists ε>0 such that for every n_0, |x-x_n|>=ε for some n>=n_0. Thus|x-x_n|>=ε for an infinite number of n . This implies that there exists a subsequence y_n of x_n , such that for each n, |x-y_n|>=ε. However the latter contradicts the fact that y_n has a subsequence that converges to x .

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Please use MathJax in your answers. – kamil09875 Jan 3 at 11:26

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