I wonder how you thought that it was not at research level!!! I am sitting here in office and it is funny to see such comments. Of course I have simplified the problem to be understood by everyone, but this is part of the problem I need to solve for a bigger proof.

Thank you a lot! This was actually a simplified version, I go back with this insight to the original one and see how I can proceed.

@ChristianRemling Thank you! I also convinced myself in another way: Since the exact solution is convex (for $\gamma,t<1$), I can show that the numerical solution cannot be bigger than the exact one; so, the boundedness of the latter is enough.

@ChristianRemling I would be happy if you are right, but I have some doubt; let me put it this way: Consider $y'=\gamma y^2$ with $\gamma<1$. I want to use forward Euler to integrate numerically from 0 to 1 with very small time step (to mimic $N\to \infty$). Can I be sure that my numerical solution stays bounded and/or close to the bounded exact solution? I have seen a bound in Burden-Faires which depends on the Lip-constant $L$ of $f=\gamma y^2$. But $L$ can be very large if the numerical solution gets unbounded and I could go nowhere.

@ChristianRemling Thank you so much! I had not thought about it this way :) It remains to validate that the difference between the numerical solution and the continuous one is bounded which I think is doable since I know the regularity of the exact solution (for $\gamma<1$).