Prove a sequence is convergent and find its limit

Question

I tried to prove that $a_{1}=s,\ a_{n+1}=s+a_{n}^{2}$ is a monotonically increasing series, but I didn't know how to prove that it is bounded from above. about the limit, I tried to compare between the limit of $a_n$ and the limit of $a_{n+1}$ but I received: $L= S + L^2$ and I didn't know how to move on with it s is between 0 to 0.25 included***

Answer 1

If the limit $L$ does exist, $L = s + L^2$ is a quadratic equation in $L$. That has at most two real roots. If there are no real roots, there is no possibility of convergence. If there are, the next step might be to look at a cobweb plot of the function $f(x) = s + x^2$.