= Solution
Fix $R>0$. Part i, with $\epsilon$ arbitrarily small and $t$ sufficiently large, shows that
$$
\mathbb P\bigl(\{z\in\mathbb H:|z|<R\}\subseteq K_t
\text{ for some }t\bigr)=1.
$$
The hulls increase with time. Once this half-disc lies in $K_t$, the stated future-avoidance property gives
$$
\gamma((t,\infty))\subseteq\overline{\mathbb H}\setminus K_t
\subseteq\{z:|z|\geq R\}.
$$
Intersecting these probability-one events over positive integer $R$ proves
$$
\liminf_{t\to\infty}|\gamma(t)|=\infty
$$
almost surely. This is <Transience of chordal SLE>.
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