= Polynomial recurrence for the absolute value
{title2=$p_{r+1}(x)=p_r(x)+(x^2-p_r(x)^2)/2$}
Starting with $p_0=0$ on $[-1,1]$, this <polynomial> recurrence gives $0\le p_r(x)\le |x|$ and $p_{r+1}(x)\ge p_r(x)$. Indeed the error satisfies $|x|-p_{r+1}=(|x|-p_r)[1-(|x|+p_r)/2]$. Its pointwise limit solves $p^2=x^2$, so it is $|x|$. <Dini's theorem> gives <uniform convergence>. Each <continuous> <piecewise linear function> is an affine function plus a finite <linear combination> of $(x-t)_+=(x-t+|x-t|)/2$. Scaling and translating the recurrence therefore gives <polynomial approximation> of every such function, and their density gives the <Weierstrass approximation theorem>.
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