So I was having difficulties in a specific part of the following Theorem:
Assuming that $Y_t$ is a lazy random walk on the discrete Torus $T_N^d$ to show was: for any $u>0$ and $K \subset \subset \Bbb{Z}^d$
$lim_{N \to \infty}P[\{Y_0,...,Y_{2\lfloor uN^d \rfloor}\} \cap \phi(K)=\emptyset]=e^{-u*cap(K)}$
First make the following assumptions: Let $\alpha,\beta\in\Bbb{R}$ satisfying $2<\alpha<\beta<d$
Let $l^*=\lfloor N^\beta\rfloor+\lfloor N^\alpha\rfloor$, $l=\lfloor N^\beta\rfloor$, $L=2\lfloor uN^d\rfloor$ and $\mathcal{K}=\lfloor L/l^* \rfloor-1$
Considering the events $\mathcal{E}_k=\{\{Y_{kl^*},...,Y_{kl^*+l}\}\cap K=\emptyset\}$ we can build the following inequality:
$0 \le lim_{N \to \infty}(P[{\bigcap_{k=0}^\mathcal{K}}\mathcal{E}_k]-P[\{Y_0,...,Y_{2\lfloor uN^d\rfloor}\}\cap \phi(K)= \emptyset])\le lim_{N \to \infty}P[\bigcup_{k=0}^\mathcal{K} \bigcup_{t=kl^*+l}^{(l+1)l*}\{Y_t\in \phi(K)\}]....$
I understood the rest of the theorem without many problems; what I don't really get is:
$lim_{N \to \infty}(P[{\bigcap_{k=0}^\mathcal{K}}\mathcal{E}_k]-P[\{Y_0,...,Y_{2\lfloor uN^d\rfloor}\}\cap \phi(K)= \emptyset])\le lim_{N \to \infty}P[\bigcup_{k=0}^\mathcal{K} \bigcup_{t=kl^*+l}^{(l+1)l*}\{Y_t\in \phi(K)\}]$
If I am not mistaken I think that they first used the complementary probability to solve this :
for $\bigcap_{k=0}^\mathcal{K}\mathcal{E}_k$ the complement is
$\{\bigcap_{k=0}^\mathcal{K}\mathcal{E}_k\}^C=\{\bigcup_{k=0}^\mathcal{K}\mathcal{E}_k^C\}=\bigcup_{k=0}^\mathcal{K}\{\exists t\in\{kl^*,...,kl^*+l\}:Y_t\in \phi(K)\}=\bigcup_{k=0}^\mathcal{K}\bigcup_{t=kl^*}^{kl^*+l}\{Y_t\in \phi(K)\}$
but how do I use this in addition with the other complement to get the result I want?