Proofs on Convex Sets. 
Prove that the set $C:=\{ x : x+S_2 \subset S_1 \}$, with $S_1,S_2\subset \Bbb R^n$ is convex if $S_1$ is convex.

I understant that a vectorial space is a convex set. So $S_1$ and $S_2$ are both convex sets. But I do not understand how to continue with the idea.

Suppose $x, y \in C$ and $0 \le t \le 1$.  You want to show $tx + (1-t)y \in C$, i.e. $tx + (1-t)y + S_2 \subset S_1$.
If $s_2 \in S_2$, $tx + (1-t) y + s_2 = t (x + s_2) + (1-t) (y + s_2)$ with
$x + s_2 \in S_1$ and $y + s_2 \in S_1$.  Since $S_1$ is convex, you are done.
