are also included in the set $E_{r-1/2}(x)$, and one of these balls is centered at a distance at most $1/2$ from $\partial
  S(x,r-1/2)$. This gives $$\label{incleq}D_r(x)\cap \{\delta_r(x)<1/2\}\subset
    D_{r-1/2}(x)\cap \{\delta_{r-1/2}(x)\ge 1/2\}.$$

We will now proceed by contradiction. Suppose that $\P[D(x)]>0$ and that $\lim_{r\rightarrow \infty}\P[\delta_{r}(x)<1/2|D_r(x)]=1.$ These assumptions imply that $$\begin{gathered}
\lim_{r\rightarrow
      \infty}\P[D_r(x)\cap
    \{\delta_r(x)<1/2\}]\\=\lim_{r\rightarrow\infty}\P[\delta_{r}(x)<1/2|D_r(x)]\P[D_r(x)]=\lim_{r\rightarrow\infty}\P[D_r(x)]=\P[D(x)]>0.\end{gathered}$$ However, by (\[incleq\]) we get that $$\begin{gathered}
\limsup_{r\rightarrow
      \infty}\P[\delta_{r-1/2}(x)\ge
    1/2|D_{r-1/2}(x)]\ge\limsup_{r\rightarrow
      \infty}\P[D_{r-1/2}(x)\cap\{\delta_{r-1/2}(x)\ge
    1/2\}]\\\ge\lim_{r\rightarrow\infty}\P[D_r(x)\cap
    \{\delta_r(x)<1/2\}]>0,\end{gathered}$$ so that in particular $\P[\delta_r(x)\ge 1/2|D_r(x)]$ does not go to $0$ as $r\rightarrow
  \infty$ which contradicts the assumption $\lim_{r\rightarrow
    \infty}\P[\delta_{r}(x)<1/2|D_r(x)]=1.$ Thus we conclude that $\P[D(x)]=0$ or/and $\liminf_{r\rightarrow
    \infty}\P[\delta_r(x)<1/2|D_r(x)]<1.$ We now assume $\liminf_{r\rightarrow \infty}\P[\delta_r(x)<1/2|D_r(x)]<1$ and show that this implies $\P[D(x)]=0$. By the assumption, we may pick a constant $c_1>0$ and a sequence of positive numbers $\{a_k\}_{k=1}^{\infty}$ such that $a_{k+1}-a_k\ge 2R+1$ and $\P[\delta_{a_k}(x)\ge 1/2|D_{a_k}(x)]\ge c_1$ for all $k$. On the event $D_{a_k}(x)$ we may pick a point $Y$ on $\partial
  S(x,a_k+R+1)$ such that if $S(Y,R+\max(0,1-\delta_{a_k}(x)))$ is completely covered by balls centered within $S(Y,R)$, then $D_{a_{k+1}}(x)^c$ occurs since a ball of radius $R$ has been found in $C(x,S(x,a_{k+1}))$ (this ball is contained in $C(x,S(x,a_{k+1}))$ since $a_{k+1}-a_k\ge 2R+1$ and $R>1$). The configuration of balls within $S(Y,R)$ is independent of the Poisson process within $S(x,a_k+1)$. Now let $\Delta_k$ be a random 