Define a sequence of polynomials inductively as $$2p_{n+1}(t)= t^2+2p_n(t)-p_n(t)^2$$ for $n \ge 0$ and where $p_0(t)=0$.Prove that $p_n(t)$ converges uniformly to $|t|$ for $t \in [-1,1]$.
I thought of using Dini's theorem, for that I need to show $p_n$ is monotonic and that would imply that $p_n(t)$ is bounded by $|t|$. But I could not prove that it is monotone.
Write $a=p_n(t)$ and $b=p_{n+1}(t)$. We can inductively assume that $0\le a\le |t|\le1$. Then $$2|t|-2b=2|t|-t^2-2a+a^2=(1-a)^2-(1-|t|)^2=(|t|-a)(2-a-|t|)\ge0$$ and $$2b-2a=t^2-a^2=(t-a)(t+a)\ge0$$ so $0\le b\le |t|$ also.