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.

Here is another way of looking at it:
Note that $C = \{ x | x +y \in S_1 \ \forall y \in S_2 \} = \{ x | x \in S_1-\{y\} \ \forall y \in S_2 \} = \cap_{ y \in S_2} (S_1 - \{y\})$.
Each set $S_1 - \{y\}$ is convex since it is the translate of a convex
set and  since the
intersection of convex sets is convex, we see that $C$ is convex.
