(zn) and (wn) if necessary, it may be assumed that zn and wn both lie in Vn. But this gives a contradiction since |f(zn) − f(wn)| ≥ ε. So f must be uniformly continuous on U. Thus f extends continuously to the closure of D. Since f(D) = U, by compactness f carries the closure of D onto the closure of U and hence ∂D onto ∂U. If f is not one-one, there are points u, v on ∂D with u ≠ v and f(u) = f(v). Let X and Y be the radial lines from 0 to u and v. Then f(X ∪ Y) is a Jordan curve. Arguing as before, its interior V is contained in U and is a connected component of U \ f(X ∪ Y). On the other hand, D \ (X ∪ Y) is the disjoint union of two open sectors W1 and W2. Hence, for one of them, W1 say, f(W1) = V. Let Z be the portion of ∂W1 on the unit circle, so that Z is a closed arc and f(Z) is a subset of both ∂U and the closure of V. But their intersection is a single point and hence f is constant on Z. By the Schwarz reflection principle, f can be analytically continued by conformal reflection across the circular arc. Since non-constant holomorphic functions have isolated zeros, this forces f to be constant, a contradiction. So f is one-one and hence a homeomorphism on the closure of D. Two different proofs of Carathéodory's theorem are described in and . The first proof follows Carathéodory's original method of proof from 1913 using properties of Lebesgue measure on the circle: the continuous extension of the inverse function g of f to ∂U is justified by Fatou's theorem on the boundary behaviour of bounded harmonic functions on the unit disk. The second proof is based on the method of , where a sharpening of the maximum modulus inequality was established for bounded holomorphic functions h defined on a bounded domain V: if a lies in V, then |h(a)| ≤ mt ⋅ M1 − t, where 0 ≤ t ≤ 1, M is maximum modulus of h for sequential limits on ∂U and m is the maximum modulus of h for sequential limits on ∂U lying in a sector centred on a subtending an angle 2πt at a. Continuous extension and the