Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 0 down vote favorite

Let $\{u^k\}\subset \mathbb{R}^n$ be a sequence satisfying 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 $\bar{u}$.

share|improve this question
3  
A Cauchy sequence converges if and only if it has a convergent subsequence. – Arturo Magidin Jul 17 '12 at 5:56
Dear Arturo Magidin. Since $\{u^k\}\subset\mathbb{R}^n$, $\{u^k\}$ is a Cauchy sequence iff it is convergent. – blindman Jul 17 '12 at 6:02
Are you asking that, if a sequence has a convergent subsequence, then the main sequence converges? – Mhenni Benghorbal Jul 17 '12 at 6:09
@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. – blindman Jul 17 '12 at 6:16

1 Answer

up vote 2 down vote accepted

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.

share|improve this answer
in our case $\{u^k\}\subset \mathbb{R}^n$, we don't consider in a general metric space. – blindman Jul 17 '12 at 6:33
Yes, but since $\mathbb{R}^n$ is in fact a metric space, the result applies. – KReiser Jul 17 '12 at 6:39
Yes, R^n is a metric space. – Mhenni Benghorbal Jul 17 '12 at 6:45
@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. – blindman Jul 17 '12 at 6:54
2  
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? – KReiser Jul 17 '12 at 8:11
show 2 more comments

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.