 Solution

The origin belongs to $S$, so $S$ is nonempty. Each displayed inequality defines a closed half-space, and every half-space is a convex set. An intersection of convex sets is convex: if $x,y\in S$ and $0\le t\le 1$, each linear inequality remains valid for $(1-t)x+ty$. Thus $S$ is a nonempty convex polytope.