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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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    @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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    @Mhenni Benghorbal, Thank you for your comments and your consideration of my question.2012-07-17