SLE reaches every positive height
= SLE reaches every positive height
{c}
For a chordal <SLE> trace $\gamma$, the <Scaling invariance of SLE> makes $\mathbb P(\sup_t\operatorname{Im}\gamma(t)>r)$ independent of $r>0$. The trace must enter the upper half-plane because its positive-time hull has positive <half-plane capacity>. Letting $r\downarrow0$ therefore makes this probability one, and continuity implies that every positive height is reached. This argument does not assume <Transience of chordal SLE>.