If A is a subset of R, let L be the set of all limit points. We want to show that L is closed. Here's how I was thinking about proving it:
To show that L is closed, we can show that the complement of L (call it c(L)) is open. c(L) is open if we can put an epsilon-neighborhood around each of the isolated points of L and always have it included in L. I'm not exactly sure where to go from here.
Any hints would be much appreciated.