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The Scaling invariance of SLE says that $r^{-1}\gamma (r^{2}t)$ has the same law as $\gamma (t)$. Therefore its first time reaching height one is $r^{-2}{\sigma }_r$, andfor every $r>0$.

To identify $p$, it suffices to look at small heights. The trace must have a point of positive imaginary part: otherwise it lies entirely on the real axis and its hull in $\mathbb{H}$ is empty, contradicting $\mathrm{hcap}(A_t)=2t>0$. By continuity from $\gamma (0)=0$, any positive height attained by the trace forces it to pass through every smaller positive height.

Consequently the increasing union of events $\{{\sigma }_{1/n}<\infty \}$ has probability one. Each event has probability $p$, so continuity of probability gives $p=1$. This proves that SLE reaches every positive height:Equivalently, a positive random height supremum whose law is invariant under every dilation must be infinite almost surely. This argument uses neither Transience of chordal SLE nor any assertion about where the trace goes as time tends to infinity.