Numerical Methods: Euler to Runge–Kutta · Superchallenge problem

The equation $y' = -y^3$ drives every solution to $0$. Euler's method with step $h = 0.02$ gives the map $y_{n+1} = y_n - 0.02\,y_n^3$. Find the supremum of the starting values $y_0 > 0$ for which the Euler iterates converge to $0$.