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Ok, I'm studying real analysis and I'm trying to make my homework, but there is a question about real analysis which I can't solve.

If $A \subset \mathbb{R}^n$ is an open and convex set then $A = \operatorname{int}(\,\overline{\!A})$

I've proved that $A \subset \operatorname{int}(\,\overline{\!A})$, because $A$ is an open set. For the other way, I guess I need to use that $A$ is a convex set to prove that $\operatorname{int}(\,\overline{\!A}) \subset A$, but I dunno how to use that assumption.

Can you help me?

Bernard
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Joãonani
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  • Take a point in $x\in B=\operatorname{int}(\overline{A})$. Enclose it in the interior of a closed box with vertices in $\overline{A}$. Find points of $A$ so close to the vertices of the box, that $x$ is in the convex hull of those points of $A$. It follows that $x\in A$. – conditionalMethod Dec 04 '19 at 19:58
  • I didn't get it, there is another way? – Joãonani Dec 04 '19 at 20:06

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