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I have a problem about proving the following. Can you please help me?

First of all, boundary of A is the set of points that for every r>0 we can find a ball B(x,r) such that B contains points from both A and outside of A.

Secondly, definition of closure of A is the intersection of all closed sets containing A. I am trying to prove that, Let A is a subset of X and X is a metric space. Closure of A = (A union boundary of A). To prove it, i try to show LHS is a subset of RHS, and then RHS is subset of LHS. It is obvious for me that RHS is subset of LHS. But why is closure of A is subset of (A union boundary A) ? For example, let x be an element in closure of A, and let us say closure of A is B. Then, if x is in B, it can be in the set B\A, i think it does not have to be in (A union boundary of A). So, i dont get it. Can anyone help?

Thanks.

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    What is your definition for the boundary of a set?2012-12-03
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    boundary of A is the set of points that for every r>0 we can find a ball B(x,r) such that B contains points from both A and outside of A.2012-12-03
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    @Magpie: its his 7th question.2012-12-04

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