how can you prove this theorem from the book?
A finite set has no accumulation points.
I know that it's true, but fail to show it mathematically
Well, if you construct an epsilon neighborhood around a proposed limit point x in your finite set, then the intersection of your set with the epsilon neighborhood of x must contain points in your finite set other than x for x to be a limit point (accumulation point). If you can show this doesn't hold, then you are done.
The definition of "accumulation point" is: p is an accumulation point of set A if and only if every neighborhood of p contains at least one point of A (other than p itself). Suppose A is finite. Then the set of all distances from p to points in A (other that p itself) is finite and so contains a smallest value. Take the radius of your neighborhood to be smaller than that value.
(I notice now, that is pretty much what Danneedshelp said!)
I am not a 100% this correct, but I recall a lemma from my intro to real analysis book that stated something along to lines of: a point is a limit point (accumulation point) of a set iff for every , the neighborhood contains infinitly many points of the set .
I think you would have to prove this to use it, but it answers your original question rather quickly.