Let $T$ denote the random time at which the process stops. Note that $T<\infty$ a.s., since if $Z^{'\lambda_2}(x)$ is unbounded, then $Z^{'\lambda_2}(x)$ contains balls of radius $R$ a.s. by Lemma \[ballemma\]. Let $F_1$ be the event that the process stops when a ball of radius $R$ is found, and note that $Z^{'\lambda_2}(x)$ is a.s. bounded on $F_1^c$. On $F_1$, we may (in some way independent of $Z^{''}$) pick a point $Y$ such that $S(Y,R)$ is covered by $Z^{'\lambda_2}[L_T^{'\lambda_2}(x)]$.

For $i=1,2$ let $$X^{\lambda_i}:=(X^{'\lambda_i}\cap
L_{T}^{'\lambda_2}(x))\cup(X^{''\lambda_i}\cap
L_T^{'\lambda_2}(x)^c)$$ and $Z^{\lambda_i}:=\cup_{x\in
X^{\lambda_i}}S(x,1)$. In this way, $Z^{\lambda_i}$ is a Poisson Boolean model with intensity $\lambda_i$ for $i=1,2$, and any ball present in $Z^{\lambda_1}$ is also present in $Z^{\lambda_2}$.

Now put $$F_2:=F_1\cap \{S(Y,R)\mbox{ intersects an unbounded
  component of }Z^{''\lambda_1}\}.$$

But on $F_2$ there is some point in $Z^{\lambda_2}(x)$ which is at distance less than or equal to two from some unbounded $Z^{\lambda_1}$ component, that is $\{\tilde{D}(x)\le2\}$ occurs for $Z$ so that $E_2(x)$ does not occur for $Z$. Since $E_2(x)$ is up to a set of measure $0$ contained in $F_1$ we have that $$\P[E_2(x)]\le\P[F_1\cap F_2^c].$$ Since $Z^{'}$ and $Z^{''}$ are independent it follows that $$\P[F_2|F_1]=\P[S(Y,R)\mbox{ intersects an unbounded component of
}Z^{''\lambda_1}]\ge 1-a$$ and consequently $$\P[F_1\cap
F_2^c]\le\P[F_2^c|F_1]<a.$$ Since 