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 :(

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- Sep 23rd 2010, 08:10 PMEmmWalferaccumulation points
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 :( - Sep 24th 2010, 05:13 AMAckbeet
Are you working in the real numbers? What ideas have you had so far?

- Sep 24th 2010, 10:03 AMDanneedshelp

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. - Sep 25th 2010, 04:26 AMHallsofIvy
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!) - Sep 25th 2010, 08:44 AMDanneedshelp
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 $\displaystyle x$ is a limit point (accumulation point) of a set $\displaystyle A$ iff for every $\displaystyle \epsilon>0$, the neighborhood $\displaystyle N_{\epsilon}(x)$ contains infinitly many points of the set $\displaystyle A$.

I think you would have to prove this to use it, but it answers your original question rather quickly.