I am reading Mathematical Analysis: An Introduction by Andrew Browder. In section 7.1, the author gave a proof of the real version of Weierstrass's Approximation Theorem, which I think is conceptually much simpler than the conventional proof based on Bernstein polynomials.
Essentially, the author first reduced the theorem to the special case where the function to be approximated is $|x|$ on $[-1,1]$. Then he constructed a sequence of polynomials $\{q_n\}_{n\in\mathbb N}$ that decreases to $|x|$ pointwise, so that the convergence is uniform, by Dini's Theorem.
The sequence is constructed recursively by $q_0=1$ and $q_{n+1}=\frac12(x^2+2q_n-q_n^2)$. It follows from mathematical induction that $1\geq q_n\geq q_{n+1}\geq|x|$ because
- $q_n-q_{n+1}=\frac12(q_n^2-x^2)\geq0$ and
- $q_{n+1}-|x|=\frac12[(1-|x|)^2-(1-q_n)^2]\geq0$.
Since $\{q_n\}$ is a decreasing sequence that is bounded below by $|x|$, it converges to some function $f$. Hence $f$ is a nonnegative solution to the equation $f=\frac12(x^2+2f-f^2)$. Therefore $f(x)=|x|$.
I like this proof. However, I don't understand how the author came up with the construction of $\{q_n\}$ above. That is, given a polynomial $p$ that satisfies $1\geq p\geq|x|$ on some interval, how can one envision that the inequality $1\geq p\geq q\geq|x|$ is satisfied by the polynomial $q=\frac12(x^2+2p-p^2)$?
(I have seen another construction in another question, but I don't see any apparent relation between the two constructions.)