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I know how to start but I don't know how to end that proof. It's supposed to be easy:

Let S a subset of Rn.

PROVE THAT the boundary of S is a closed set.

(I'll use d for delta, so dS is my convention for "the boundary of S").

So here I go:

dS is closed iff it contains all of its boundary points,

so dS is closed iff d(dS) is included in dS.

Let x be any point such that x belongs to d(dS).

So for any Ball B(r, x), r>0, (ie centered at x),

| B intersection dS is not empty

| and B interesection (dS)complement is not empty.

(the second line is equivalent to) B interesection (interiorOfS union exteriorOfS) is not empty

Now what's next??

Thanks for your suggestions. If you do have a suggestion, please don't skip a step or don't bypass a detail because it seems obvious (trust me, nothing is obvious to the one who doesn't know yet!)