Universal Taylor series

Let [math]\displaystyle{ f_1, f_2, ... }[/math] be the sequence in which each rational-coefficient polynomials with zero constant coefficient appears countably infinitely many times (use the diagonal enumeration). By Weierstrass approximation theorem, it is dense in [math]\displaystyle{ C[-1,1]_0 }[/math]. Thus it suffices to approximate the sequence. We construct the power series iteratively as a sequence of polynomials [math]\displaystyle{ p_1, p_2, ... }[/math], such that [math]\displaystyle{ p_n, p_{n+1} }[/math] agrees on the first [math]\displaystyle{ n }[/math] coefficients, and [math]\displaystyle{ \|f_n - p_n \|_\infty \leq 1/n }[/math].

To start, let [math]\displaystyle{ p_1 = f_1 }[/math]. To construct [math]\displaystyle{ p_{n+1} }[/math], replace each [math]\displaystyle{ x }[/math] in [math]\displaystyle{ f_{n+1} - p_n }[/math] by a close enough approximation with lowest degree [math]\displaystyle{ \geq n+1 }[/math], using the lemma below. Now add this to [math]\displaystyle{ p_n }[/math].

The function [math]\displaystyle{ g(x) = x - c\tanh (x/c) }[/math] is the uniform limit of its Taylor expansion, which starts with degree 3. Also, [math]\displaystyle{ \|f-g\|_\infty \lt c }[/math]. Thus to [math]\displaystyle{ \epsilon }[/math]-approximate [math]\displaystyle{ f(x) =x }[/math] using a polynomial with lowest degree 3, we do so for [math]\displaystyle{ g(x) }[/math] with [math]\displaystyle{ c \lt \epsilon/2 }[/math] by truncating its Taylor expansion. Now iterate this construction by plugging in the lowest-degree-3 approximation into the Taylor expansion of [math]\displaystyle{ g(x) }[/math], obtaining an approximation of lowest degree 9, 27, 81...