I was in an other proof, in that we have to find a positive lower bound for $|z-w|,$ where where $w$ lies in the boundary of the open ball $B(a,r)$ and $z\in B(a,r/2)$.
From the figure we can easily identify that $|z-w|\geq r/2$, my try is to use the result $B(a,r)=a+B(0,r)$, But I am not convinced, Is there any easy way to prove this?