Consider the initial value problem:

y' = 1 + y^2 ; with y = 0 when x = 0.

First part of the question was to get y2(x) by successive approximation. I was able to do that.

I need help on this part:

Let R = [-1, 1] x [-1, 1]. Find the smallest M such that abs(f(x, y)) <= M on R. Find an interval I = (-c, c) such that the graph of every approximation function Yn over I will lie in R.

I'm really stuck on this problem, I don't know where to start.