What is the convex hull of $\text{conv}(u_1,u_2,\cdots,u_p)+\text{conv}(v_1,v_2,\cdots,v_s)$? Let $u_i, i= 1,\cdots,p$ and $v_j, j= 1,\cdots,s$ be finitely many vectors in $\mathbb{R}^n$. Show that 
$$
\text{conv}(u_1,u_2,\cdots,u_p)+\text{conv}(v_1,v_2,\cdots,v_s)=\text{conv}\{u_i+v_j \mid i= 1,\cdots,p, \,\, j= 1,\cdots,s\}
$$
We need to show 
$$
x+y \in \text{conv}\{u_i+v_j \mid i= 1,\cdots,p, \,\, j= 1,\cdots,s\}
$$
where $x \in \text{conv}(u_1,u_2,\cdots,u_p)$ and $y \in \text{conv}(v_1,v_2,\cdots,v_s)$. Also, we need to show 
$$
z \in \text{conv}(u_1,u_2,\cdots,u_p)+\text{conv}(v_1,v_2,\cdots,v_s)
$$
where $z \in \text{conv}\{u_i+v_j \mid i= 1,\cdots,p, \,\, j= 1,\cdots,s\}$.
I have tried the following for the first one:
Let $x \in \text{conv}(u_1,u_2,\cdots,u_p)$ so $x=\sum_{i=1}^p\lambda_iu_i$ where $\sum_{i=1}^p\lambda_i=1$. Also, Let $y \in \text{conv}(v_1,v_2,\cdots,v_s)$ so $x=\sum_{j=1}^s\mu_jv_j$ where $\sum_{j=1}^s\mu_j=1$.
Summing them 
$$x+y=\lambda_1u_1+\lambda_2u_2+\cdots+\lambda_pu_p+\mu_1v_1+\mu_2v_2+\cdots+\mu_sv_s.$$
Now the question is how we can get something in the form of $\text{conv}\{u_i+v_j \mid i= 1,\cdots,p, \,\, j= 1,\cdots,s\}$?

For two sets
$$\operatorname{co}(S_1 \times S_2) = \operatorname{co}(S_1) \times \operatorname{co}(S_2)$$
the inclusion from left to right is clear, for the other way notice
the equality
$$(\sum \lambda_i u_i, \sum \mu_j v_j) = \sum \lambda_i \mu_j (u_i , v_j)$$
if $\lambda_i$, $\mu_j$ positive with sum $1$. Now apply the affine map $+\colon V\times V \to V$.
