Topological Analog of the Weierstrass Double Series Theorem

Since any light interior transformation of a sphere or a Riemann surface into a sphere is topologically equivalent to an analytic transformation and any non-constant analytic transformation is light and interior, it might be expected in view of the Weierstrass Double Series Theorem that the limit of a uniformly convergent sequence of light interior mappings of a sphere into a sphere would be light and interior. However, this is readily seen not to be the case. For if we let r denote |0| and for each w>0we map the complex sphere onto itself by the function fn(z) defined by