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Recursive sequence and a quadratic equation related inequality proof

I am trying to show that if a sequence of number $x_{n}$ is defined by $x_1 = h$, $x_{n+1}=x_n^2 + k$, where $0<k<\frac{1}{4}$ and $h$ lies between the roots $a$ and $b$ of the equation $$x^2 -x ...