Suppose that I have the following recursion for all $a_k >1$: $a_{k+1}^2 \leq a_k^2 - a_k$. Then one can see that the following inequality is true (proved via induction): $a_k \leq a_0 - \frac{k}{2}$. Now if I update my recursion as $a_{k+1}^2 \leq a_k^2 - a_k +\frac{1}{2}$, the inequality, $a_k \leq a_0 - \frac{k}{2}$, does not hold anymore.

I was wondering if one can show a modifed inequality of the form $a_k \leq a_0 - \frac{k^\alpha}{b}+c$, for some value of $\alpha,b,c$ holds true. But it seems to involve conditions on $a_0$ which seems unsatisfactory.

Can we show an upper bound on $a_k$ with the second recursion in terms of $a_0$ and some function of $k$ ?