Universal Taylor series

Let ${\displaystyle f_1,f_2,...}$ 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 ${\displaystyle C[-1,1{]}_0}$. Thus it suffices to approximate the sequence. We construct the power series iteratively as a sequence of polynomials ${\displaystyle p_1,p_2,...}$, such that ${\displaystyle p_n,p_{n+1}}$ agrees on the first ${\displaystyle n}$ coefficients, and ${\displaystyle \Vert f_n-p_n{\Vert }_{\mathrm{\infty }}\le 1/n}$.

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

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