Consider a real sequence $(x\_k)$ for $k=0,1,2,\dots,N$ as $x\_0=1$ and for $k>0$
$$ x\_k=x\_{k-1}+\frac{\gamma}{N}x\_{k-1}^2,\qquad (\gamma>0).$$
I wonder to show that the sequence is bounded as $N\to\infty$. I appreciate any idea for proving that.


Hint 1: Numerical experiments suggest me that if and only if $\gamma<1$ the sequence is bounded.


Hint 2: It is known that for $x\_k=x\_{k-1}+\frac{\gamma}{N}x\_{k-1}$ **which is linear**, a bound exists as $e^{\gamma}$. 



boundedness of a nonlinear recursive sequence
You can view this difference equation as the Euler method for the IVP $y'=\gamma y^2$, $y(0)=1$, on the interval $0\le t\le 1$, using a grid of width $1/N$ and setting $x\_k=y(k/N)$.


By solving the ODE, we find that $y$ blows up at $t=1/\gamma$. We want to know if $x\_N=y(1)$ stays bounded, and it now follows that this will be the case if $\gamma<1$ (as you suspected), while $x\_N$ becomes unbounded if $\gamma>1$. The borderline case $\gamma=1$ would seem to require a more careful analysis.


