I am trying to construct a sequence to show that:

Let $(V,||.||)$ be a normed vector space. Then if $S={vin V: ||v||=1}$ 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.