Prove a sequence is convergent and find its limit 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***

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 $f(x) = s + x^2$.
