TITLE: How to prove that if convex $A \subset \mathbb{R}^n$ and $A+A=A$ then $0 \in cl(A)$?
QUESTION [7 upvotes]: How to prove that if convex $A \subset \mathbb{R}^n$ and $A + A = A$ then $0 \in cl(A)$? 
For a example of $A$ which holds a condition but $0 \notin A$ consider $(0,\infty)$.

REPLY [7 votes]: Assume $A\neq\emptyset$. Let $x\in A$. Then since $A+A\supset A$ there exists $y,z\in A$ such that $x=y+z$. But then by convexity $\tfrac{1}{2}x = \tfrac{1}{2}(y+z)\in A$. Repeating this argument with $\tfrac{1}{2}x$, $\tfrac{1}{4}x$, and so on, you get $\tfrac{1}{2^k} x\in A$. Letting $k\to\infty$, $0\in\text{cl}(A)$.