Best uniform polynomial error on a compact interval

Let $I=[a,b]\subset \mathbb{R}$ with $a<b$, and let $C(I,\mathbb{R})$ denote the real-valued continuous functions on $I$. For $h\in C(I,\mathbb{R})$ set $\mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty },I}:={\sup }_{x\in I}\mathrm{\mid }h(x)\mathrm{\mid }$. For $n\in {\mathbb{N}}_0$, let ${\mathcal{P}}_n(I)$ be the restrictions to $I$ of real polynomials of degree at most $n$, including the zero polynomial, and define $E_n(f;I):={\inf }_{p\in {\mathcal{P}}_n(I)}\mathrm{\parallel }f-p{\mathrm{\parallel }}_{\mathrm{\infty },I}\mathrm{.}$ When $I=[-1,1]$, this agrees with the published definition Best uniform polynomial error and Chebyshev interpolation operators.

These quantities are finite and well-defined. The interval is compact by For a nonempty subset of ${\mathbb{R}}^n$ with $n\ge 1$, compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent, so each continuous difference $f-p$ has a finite maximum absolute value. The polynomial space contains $0$, hence the set of errors is nonempty and bounded below by $0$; completeness of $\mathbb{R}$ (Complete ordered field (least-upper-bound property)) supplies its infimum.