I am having a hard time with a proof:
Prove that the set of limit points of a set is closed.
At first it seemed easy and from a logical standpoint it makes sense but I am not sure where to start.
what definition are you using for a "limit point"? a "closed set"?
strategies you may want to explore:
- show that the compliment of the set of limit points is open (this is perhaps the easiest way)
- show that if S is the set of limit points, it will contain all the limits of its convergent sequences (consisting of points in S)
- recall also that x is a limit point of a set S if it lies in the closure of S - {x}. And show that this set is closed. (this amounts to showing that the closure of a set is closed, i think)
- you can also show that for any point in S (the set of limit points of a set) that has the property that any open ball centered at that point intersects S and S compliment (that is, it is a boundary point of S) will be contained in S. (we know that such points exists based on one of the definitions for a limit point)*. thus we would show that S contains all its boundary points and is hence closed
*) Definition: Let S be a subset of a topological space X. We say that a point x in X is a limit point of S if every open set containing x also contains a point of S other than x itself.