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Let $\{u^k\}\subset \mathbb{R}^n$ be a sequence such that there exists a subsequence $\{u^{k_i}\}\subset \{u^k\}$ converging to $\bar{u}\in \mathbb{R}^n$.

I would like to ask when we have a stronger conclusion that $\{u^k\}$ converges to $\bar{u}$. For example, if $\{\|u^k-\bar{u}\|\}$ is monotonically decreasing then $\{u^k\}$ converges to $\bar{u}$.

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    A Cauchy sequence converges if and only if it has a convergent subsequence.2012-07-17
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    Dear Arturo Magidin. Since $\{u^k\}\subset\mathbb{R}^n$, $\{u^k\}$ is a Cauchy sequence iff it is convergent.2012-07-17
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    Are you asking that, if a sequence has a convergent subsequence, then the main sequence converges?2012-07-17
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    @Mhenni Benghobal. I would like to find some sufficient conditions for the sequence $\{u^k\}$ to be convergent in the case $\{u^k\}$ has a convergent subsequence.2012-07-17

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You need to assume that the main sequence is Cauchy, and by the following lemma, your sequence converges:

Let $(X,d)$ be a metric space, and let $(a_k)$ be a Cauchy sequence in $X$. Then $(a_k)$ converges iff $(a_k)$ has a convergent subsequence.

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    in our case $\{u^k\}\subset \mathbb{R}^n$, we don't consider in a general metric space.2012-07-17
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    Yes, but since $\mathbb{R}^n$ is in fact a metric space, the result applies.2012-07-17
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    Yes, R^n is a metric space.2012-07-17
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    @KReiser, I understand completely what you mentioned, but I would like to find more properties of the sequence $\{u^k\}\subset \mathbb{R}^n$ to guarantee $\{u^k\}$ to be convegent in the case it has a convergent subsequence.2012-07-17
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    Any supposed technique that will show the result you ask for will necessarily prove that the sequence is Cauchy. Under most circumstances, checking that a sequence is Cauchy is usually easy enough to do. Perhaps if you have a specific example in mind where the check is hard, you could post it?2012-07-17
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    @KReiser, Thank you for all your comments and your consideration of my question.2012-07-17
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    @Mhenni Benghorbal, Thank you for your comments and your consideration of my question.2012-07-17