= Solution
The <uniformization theorem> says that every simply connected <Riemann surfaces>[Riemann surface] is conformally equivalent to exactly one of the <Riemann sphere>, the <complex plane>, and the <unit disc>. A proper simply connected plane domain is noncompact, so it is not the sphere. It cannot be uniformized by the plane: after choosing a point outside the domain, a branch of a square root and then a <Möbius transformation> would produce a nonconstant bounded entire function, contradicting the <Liouville theorem>. It is therefore conformally equivalent to the disc, which is the <Riemann mapping theorem>.
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