Solution
= Solution
A closed circuit of diameter at least $n$ through a fixed listed point contains a closed path from that point to graph distance at least $n/2$. <Exponential decay of subcritical percolation> bounds this probability by $Ce^{-cn}$. A union bound over the $n$ listed points gives
$$
\mathbb P(B_n)\leq Cn e^{-cn}\leq n e^{-u n}
$$
after reducing $u>0$ and adjusting finitely many small $n$.