SLE reaches every positive height

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