= Solution
The events $A_n$ decrease, and an open path from $0$ reaches every $\partial\Lambda_n$ exactly when the open cluster of $0$ is infinite. Therefore <continuity from above of a measure> gives
$$
\mathbb P_p(A_n)\downarrow\mathbb P_p(0\leftrightarrow\infty)=\theta(p).
$$
Each $A_n$ depends on finitely many sites, so $p\mapsto\mathbb P_p(A_n)$ is a polynomial and hence <continuous function>[continuous]. Given $\varepsilon>0$, choose $n$ with $\mathbb P_p(A_n)<\theta(p)+\varepsilon$. For $p'\downarrow p$, monotonicity and the finite-event continuity yield
$$
\theta(p)\leq\theta(p')\leq\mathbb P_{p'}(A_n)\longrightarrow\mathbb P_p(A_n)<\theta(p)+\varepsilon.
$$
Thus $\theta(p')\to\theta(p)$ from the right.
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