= Cosine fixed-point iteration
{title2=$x_{n+1}=\cos x_n$}
Use radians and take $T(x)=\cos x$ on the <closed interval> $[0,1]$. This is a <complete metric space> with distance $|x-y|$, and $T([0,1])=[\cos1,1]\subset[0,1]$. The <mean value theorem> and $|T'(x)|=\sin x\leq\sin1$ give a <contraction mapping> with $L=\sin1\approx0.841471<1$. Both the interval and its preservation matter; a small <derivative> by itself would not ensure that all subsequent approximations stay in the chosen space.
Starting at $x_0=0$ gives $x_1=1$, $x_2\approx0.540302$, $x_3\approx0.857553$, and $x_4\approx0.654290$. The <cosine> is decreasing here, so successive iterates alternate around the unique <fixed point> $x_*\approx0.739085133215161$. The <contraction iteration error bound> needs no prior knowledge of this numerical value. At step 40, $x_{40}\approx0.739085078689123$ and the latest-step bound is below $7.20\times10^{-7}$, certifying error below one millionth for exact iteration. The initial-step bound requires 91 iterations for the same tolerance. The observed step-40 error is only about $5.45\times10^{-8}$; the guarantees allow less favorable behavior than this example actually exhibits.
\Image[/contraction-cosine-iteration.png]
{title=Cosine iteration converges with computable error bounds}
{description=The staircase alternates around the intersection of $y=\cos x$ and $y=x$. The error plot compares measured error against the initial-step and latest-step contraction bounds; both bounds hold without knowing the fixed point.}
This is the beauty of the <contraction mapping theorem>: a uniform local improvement controls the entire infinite process. <Completeness> is essential. On the incomplete interval $(0,1)$, the <contraction mapping> $T(x)=x/2$ keeps every iterate inside but has no <fixed point> there; its missing limit is zero. Banach's 1922 paper developed abstract methods for different spaces of functions with applications to <integral equations>, a motivation still visible whenever repeated approximation is used to construct a function.https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/3/0/92453/sur-les-operations-dans-les-ensembles-abstraits-et-leur-application-aux-equations-integrales[Stefan Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundamenta Mathematicae 3 (1922), 133–181]{ref}
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