 Solution

It suffices to use the two-dimensional coordinate half-plane inside $H$. A Peierls argument shows that when $p$ is sufficiently close to one, the probability that a fixed open vertex is separated from distance $n$ by a closed dual contour is summable over contour lengths: the number of length-$m$ contours is at most exponential in $m$, whereas each is closed with probability $(1-p{)}^m$. Hence the origin has positive probability of belonging to an infinite open cluster. The event that some infinite cluster exists is invariant and therefore has probability zero or one because the boundary translation is an ergodic transformation; its positive probability makes it almost sure. The same cluster is also an infinite cluster of $H$ in every $d\ge 2$.