Constructing a bounded sequence with no convergent subsequence

Asked Sep 14 '20 at 06:38

Active Sep 14 '20 at 07:07

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I am trying to construct a sequence to show that:

Let $(V,||.||)$ be a normed vector space. Then if $S=\{v\in V: ||v||=1\}$ is not compact there is a bounded sequence in V which has no convergent subsequence.

If S is not compact, does it follow that there exists $ϵ$ such that ${B(x,ϵ):x∈S}$ has no finite subcover? I'm pretty sure I can construct a sequence with no Cauchy subsequence if this is the case, but I do not know how to justify it.

Otherwise, I would not know where to begin.

Any hints/help would be much appreciated.

asked Sep 14 '20 at 06:38

stokes

Do you know what sequential compactness is? – Kavi Rama Murthy Sep 14 '20 at 06:40
Yes @KaviRamaMurthy – stokes Sep 14 '20 at 06:45
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Take a free infinite family as your sequence. This is similar to taking $u_n=(0,...,0,1,0,...)$ in the space $\ell^2(\mathbb{N})$. – Anthony Sep 14 '20 at 06:50
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If $S$ is not compact then it is not sequentially compact which means there is a sequence in $S$ which has no convergent subsequence. There is nothing more to be done. – Kavi Rama Murthy Sep 14 '20 at 07:21

Constructing a bounded sequence with no convergent subsequence

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