Prove Convex hull is Convex
If $s_1, s_2 \in \operatorname{co} S$, then there are (a finite number of) $\lambda_k(1), \lambda_k(2)$ and $x_k \in S$ such that $s_i = \sum_k \lambda_k(i) x_k$ with $\lambda_k(i) \ge 0$ and $\sum_k \lambda_k(i) = 1$.
Now choose $t \in [0,1]$ and pick $s = ts_1 + (1-t) s_2$, then $s = \sum_k (t \lambda_k(1)+ (1-t) \lambda_k(2)) x_k$ and we can see that $t \lambda_k(1)+ (1-t) \lambda_k(2) \ge 0$ and we can quickly check that $\sum_k (t \lambda_k(1)+ (1-t) \lambda_k(2)) = 1$. Hence $s \in \operatorname{co} S$.