Let $f(x)=\frac{1}{4}+x-x^{2}$. For any real number $x$, define a sequence $\left(x_{n}\right)$ by $x_{0}=x$ and $x_{n+1}=f\left(x_{n}\right)$. If the sequence converges, let $x_{\infty}$ denote the limit. For $x=0$, find $x_{\infty}=\lambda$.


We have

$$
f(x)=\frac{1}{2}-\left(x-\frac{1}{2}\right)^{2}
$$

so $x_{n}$ is bounded by $1 / 2$ and, by the Induction Principle, nondecreasing. Let $\lambda$ be its limit. Then

$$
\lambda=\frac{1}{2}-\left(\lambda-\frac{1}{2}\right)^{2}
$$

and, as the sequence takes only positive values,

$$
\lambda=\frac{1}{2} \text {. }
$$

