Coverings by convex bodies and inscribed balls

Let $H$ be a Hilbert space. For a closed convex body $A$ denote by $r(A)$ the supremum of the radiuses of balls contained in $A$. We prove that $\sum_{n=1}^\infty r(A_n) \ge r(A)$ for every covering of a convex closed body $A \subset H$ by a sequence of convex closed bodies $A_n$, $n \in \mathbb{N}$. It looks like this fact is new even for triangles in a 2-dimensional space.