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?
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".