sequence has no zeros nor can the limit. For the second statement suppose that and set . These are nowhere-vanishing on a disk but vanishes at , so must vanish identically. Definitions. A family of holomorphic functions on an open domain is said to be normal if any sequence of functions in has a subsequence that converges to a holomorphic function uniformly on compacta. A family is compact if whenever a sequence lies in and converges uniformly to on compacta, then also lies in . A family is said to be locally bounded if their functions are uniformly bounded on each compact disk. Differentiating the Cauchy integral formula, it follows that the derivatives of a locally bounded family are also locally bounded. Montel's theorem. Every locally bounded family of holomorphic functions in a domain is normal. Let be a totally bounded sequence and chose a countable dense subset of . By locally boundedness and a "diagonal argument", a subsequence can be chosen so that is convergent at each point . It must be verified that this sequence of holomorphic functions converges on uniformly on each compactum . Take open with such that the closure of is compact and contains . Since the sequence is locally bounded, on . By compactness, if is taken small enough, finitely many open disks of radius are required to cover while remaining in . Since , we have that . Now for each choose some in where converges, take and so large to be within of its limit. Then for , Hence the sequence forms a Cauchy sequence in the uniform norm on as required. Riemann mapping theorem. If is a simply connected domain and , there is a unique conformal mapping of onto the unit disk normalized such that and . Uniqueness follows because if and satisfied the same conditions, would be a univalent holomorphic map of the unit disk with and . But by the Schwarz lemma, the univalent holomorphic maps of the unit disk onto itself are given by the Möbius transformations with . So must be the identity map and . To prove existence, take to be the family of holomorphic univalent mappings of into the open unit disk with and . It is a normal family by Montel's theorem. By the characterization of simple-connectivity, for there is a holomorphic branch of the square root in . It is univalent and