$k$ points of contact for percolation

Question

In Grimmett's book on percolation near the beginning of Ch. 7, he summarizes the plan of proof of the result on percolation of slabs. We are in $\mathbb{Z}^d, d\ge2$, with his usual notation that $B_m$ means a box of sidelength $2m$. $d$ and $p$ for which the percolation probability $\theta_p$ is positive are fixed. One claim he makes is that if $\epsilon>0$, $k\ge1$, $m\ge1$ are given for which $P_p(B_m\leftrightarrow \infty)>1-\epsilon$ then there is some $n>2m$ for which $P_p(B_m\leftrightarrow \text{ at least k distinct points in }\partial B_n)>1-2\epsilon$. I am having trouble establishing this.

Answer 1

He establishes a result that implies this in Lemma (7.9):

If $\theta(p) > 0$ and $\eta > 0$, there exists integers $m = m(p, \eta)$ and $n = n(p, \eta)$ such that $2m < n$ and $$P_p(B(m) \leftrightarrow K(m,n) \, \textrm{in}\, B(n)) > 1 - \eta$$

Is there an issue you are having with the proof of that lemma?