Scaling Limit of Successive Approximations for w′=-w²

We prove existence of scaling limits of sequences of functions defined by the recursion relation w′_n+1(ϰ)= -w_n(ϰ)². which is a successive approximation to w′(ϰ)= -w_n(ϰ)² a simplest non-linear ordinary differential equation whose solutions have moving singularities. Namely, the sequence approaches the exact solution as n→ √ in an asymptotically conformal way, [formula omitted], for a sequence of numbers {qn} and a function [formula omitted]. We also discuss implication of the results in terms of random sequential bisections of a rod.