Green-function exhaustion proof of disk uniformization

Exhaust a simply connected hyperbolic Riemann surface in potential theory by compact bordered Jordan domains $R_n$. Their Green functions on a Riemann surface $G_n(\cdot ,p)$ have logarithmic poles and zero boundary values. Exponentiating their harmonic conjugates produces proper degree-one maps $f_n:R_n\to \mathbb{D}$ with $f_n(p)=0$. If $G_n=-\log |\zeta |+c_n+o(1)$, normalize $f_n^{\prime }(p)=e^{-c_n}>0$. Domain monotonicity and a global positive Green function bound $c_n$ increasingly from above. Thus Montel theorem gives a nonconstant limit $F$, which is injective by the locally uniform limit of univalent functions theorem. The Schwarz lemma makes $F^{\prime }(p)$ extremal among normalized disk-valued maps. If $F$ omitted $a\ne 0$, taking a global holomorphic square root of a disk automorphism applied to $F$, then renormalizing at $p$, would multiply its derivative by $(1+|a|)/(2\sqrt{|a|})>1$. Thus $F$ is onto and gives the desired biholomorphism.