restricted set of a convex set

Question

Let $S \subset \mathbb R^n$, $S$ is convex and let $||.||$ be a norm on $\mathbb R^n.$ For $a \ge 0$ we define $S_{-a} =\{ x | B(x,a) \in S\}$, where $B(x,a)$ is the ball (in the norm $||.||)$, centered at $x$, with radius $a$. Prove that for any $x \in S$ and for all $u \in \mathbb R^n$ with $||u|| \le a$ then $x-u \in S_{-a}$.