Polynomial recurrence for the absolute value ($p_{r+1}(x)=p_r(x)+(x^2-p_r(x{)}^2)/2$)

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.