in $C^{\lambda}\cap A$. Let $E_r(x)$ be the union of all balls centered within $S(x,r+1)$ that are connected to $x$ via a chain of balls centered within $S(x,r+1)$. Note that $C^{\lambda}(x,S(x,r))\subset
E_r(x)$.

Let $\delta_r(x):=\sup_{y\in E_r(x)\setminus S(x,r)}d(y,\partial
S(x,r))$ where the supremum is defined to be $0$ if $E_r(x)\setminus
S(x,r)$ is the empty set. Let $\{A\leftrightarrow B\}$ be the event that there is some continuous curve in $C^{\lambda}$ which intersects both the set $A$ and the set $B$. Let $A^o$ be the interior of the set $A$.

Fix a point $x\in M$. Since the case $R\le 1$ is trivial, we suppose $R>1$. For any $r>0$ let $F_r(x):=\{x\leftrightarrow
  \partial S(x,r)\}$ and let $$G_r(x):=\{C^{\lambda}(x,S(x,r))\mbox{
    does not contain a ball of radius }R\}.$$ Let $D_r(x):=F_r(x)\cap
  G_r(x)$. Let $D(x)$ be the event that $x$ is an unbounded component that does not contain a ball of radius $R$. Then $D_r(x)\downarrow
  D(x)$ so it is enough to show that $\P[D_r(x)]\rightarrow 0$ as $r\rightarrow \infty$. Note that $D_r(x)$ is independent of the Poisson process outside $S(x,r+1)$. Also note that $\delta_{r}(x)\in[0,2]$.

If $D_r(x)\cap \{\delta_{r}(x)< 1/2\}$ occurs, then there is a ball centered in $S(x,r-1/2)^o\setminus S(x,r-1)^o$ which is connected to $x$ by a chain of balls centered in $S(x,r-1/2)^o$. All these balls 