Let $\alpha,\beta \in \mathbb{R}$ with $\alpha \leq \beta$, let $I=[\alpha,\beta]$, and let $f:I \to I$ be order-preserving, meaning that for all $x,y \in I$, if $x \leq y$, then $f(x) \leq f(y)$. Let $(a_n)_{n=1}^{\infty}$ be the fixed point iteration generated by $f$ from $a_1 \in I$, so that
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\begin{align*}
a_{n+1}=f(a_n)
\end{align*}
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for every $n \in \mathbb{N}$.
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If $a_1 \leq f(a_1)$, then $(a_n)_{n=1}^{\infty}$ is increasing in the non-strict sense and converges to some $L \in I$. If, in addition, $f$ is continuous, then $f(L)=L$.
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If $a_1 \geq f(a_1)$, then $(a_n)_{n=1}^{\infty}$ is decreasing in the non-strict sense and converges to some $L \in I$. If, in addition, $f$ is continuous, then $f(L)=L$.