Given this recurrent relation
$$a_{n+1} \leq a_n-ka_n^2,\ k>0$$
How can I prove this asymptotic behavior not using induction?
$$a_n\leq\frac{1}{nk+a_0^{-1}}$$
I ran this numerically on MATLAB and verified it for a large number of terms.
This problem comes from the third form of entropy condition for advection equations, for my study in numerical methods.
Thanks in advance.
$$a_{n} \leq a_{n-1}-ka_{n-1}^2 \iff a_n^{-1} \ge \frac{1/k}{a_{n-1}(1/k - a_{n-1)}} = a_{n-1}^{-1} + \frac{1}{1/k - a_{n-1}}$$ We deduce that $$a_n^{-1} \ge a_{n-1}^{-1} + k \ge … \ge a_0^{-1}+nk\tag{1}$$ From $(1)$, we will have easily your result.