how can one proof that a set is closed iff it contains all its limit sets.
Then the problem makes no sense. "Limits", and so "limit points, are not defined in general vector spaces. You have to have a topology to have limits. This really should be in the "Analysis, Topology, and Differential Geometry" section.
The fact that a set is closed if and only if it contains all of its limit points is a fairly standard topology question. There are, however, several different, though equivalent, definitions of "closed set". What definition are you using?
He probably meant a normed vector space $\displaystyle \left(\mathcal{V},\|\cdot\|\right)$ and the associated metric $\displaystyle d:\mathcal{V}\times\mathcal{V}\to\mathbb{R}u,v)\mapsto \|u-v\|$ and supplying $\displaystyle \mathcal{V}$ with the metric topology induced by $\displaystyle d$
Yes, I thought of that, but there was no point in working on that without the OP saying so. Also, the "vector space" properties play no role in the proof- it is a straight topology proof and would be better put in "Analysis, Topology, and Differential Geometry".