Is this proof correct?
Let $S$ be a polytope in $\mathbb R^n,$ and let $S_j=\{\mu_jd_j:\mu_j\ge 0\}$, where $d_j\in\mathbb R^n$ is different from zero and $j=1,2,...,k$.Prove that $S\oplus S_1\oplus ... \oplus S_k $ is a closed convex set.
Proof:
(in a previous exercise I proved $S_1\oplus S_2$ is closed and convex) So I have $S_1\oplus ... \oplus S_k $ is a closed convex set.
By another proved exercise I have that $S$ is a closed convex set.
The sum of convex sets is convex, so $S\oplus S_1\oplus ... \oplus S_k $ is convex.
Notice that $S=\{\sum_{j=1}^{k}\mu_jd_j\}=\{\mu_1d_1+...+\mu_k\}$ so it's elements $\mu_jd_j$ are like the elements of $S_j$. Thus $S$ can be taken as an element of this type $S_j$.
Therefore $S\oplus S_1\oplus ... \oplus S_k $ is a closed convex set.