Suppose that A and B are non-empty sets of real numbers, and that x is a limit point of A union B. Prove that x is a limit point of A or of B.

=> I think that I need to do proof by cases for this one, and use the epsilon neighborhood in some way, but I am not sure.

Also, could this be more easily proved using proof by contradiction,

assuming that x is not a limit point of a and x is not a limit point of B.

Thanks for you help.