= Solution
Assume $\theta(1/2)>0$. The event that some <infinite percolation cluster> exists is invariant under lattice translations. By <translation ergodicity of Bernoulli percolation>, its positive <probability> is therefore one.
Let $D_N$ be the event that an <infinite percolation cluster> meets the finite <graph vertex> box $B_N=[-N,N]^2$. These <increasing events> increase with $N$, and their union is the event that some <infinite percolation cluster> exists. <Continuity from below of a measure> gives
$$
\mathbb P_{1/2}(D_N)\longrightarrow1.
$$
An <infinite percolation cluster> meeting $B_N$ contains an infinite simple open <ray in a graph> from a <graph vertex> of the box. This follows from local finiteness and the <König infinity lemma>. That ray visits the finite box only finitely many times; its last box <graph vertex> lies on the boundary, and the subsequent ray is outside. Thus $D_N$ is also the event that the box boundary intersects an <infinite percolation cluster>, and is the union of the four exterior-side arm events.
Consequently, for every $\epsilon>0$, choose $N$ with
$$
\boxed{\mathbb P_{1/2}(\partial B_N\text{ meets an infinite open cluster})>1-\epsilon.}
$$
For $\epsilon\ge1$ any sufficiently large $N$ still satisfies the strict inequality. No uniqueness assertion is needed for this preliminary step.
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