Let $Y$ is vector topo space, $A\neq \emptyset, A \subset Y$ and $K$ is convex cone. Then
$$cl(A+K)+K=cl(A+K).$$
I need show that.
I have proof as follows:
For any $z \in cl(A+K)+K,$ exist ${a_{n}} \subset A,{b_{n}} \subset K, b' \in K$ such that $a_{n}+ b_{n}+ b' \to z$. Suppose that $b'_{n}:=b_{n}+b'$. Since $K + K \subset K$ so ${b'_{n}} \subset K$. Therefore, $z \in cl(A+K)$.
Is that true? Thanks for help me.