An obstacle encountered in a proof of the existence of a best approximating polynomial of degree $\leq n$

Question

Let $n \in \{0, 1, 2, \dots\}$, let $a, b \in \mathbb{R}$ be such that $a < b$ and let $f \in \mathcal{C}[a, b]$ be a real function that is continuous on the non-degenerate, compact interval $[a, b]$. Define the following two functions $D : \mathbb{R}^{n + 1} \rightarrow \left(\mathbb{R} \rightarrow \mathbb{R}\right)$ and $E : \mathbb{R}^{n + 1} \rightarrow [0, \infty)$ as follows. $$ \begin{align} D(c_0, \dots, c_n) & := x_{\in \mathbb{R}} \mapsto \sum_{k = 0}^n c_k x^k \\ E(c_0, \dots, c_n) & := \|f - D(c_0, \dots, c_n)\|_\infty \end{align} $$ where $\|\cdot\|_\infty$ is the supremum norm on $\mathcal{C}[a, b]$, i.e. for all $g \in \mathcal{C}[a, b]$ $$ \|g\|_\infty := \max_{x \in [a, b]} |g(x)| $$ It can be shown that $E$ is continuous w.r.t. the Euclidean metrics on $\mathbb{R}^{n + 1}$, $\mathbb{R}$, respectively.

Now denote by $S$ the set of all points $(c_0, \dots, c_n) \in \mathbb{R}^{n + 1}$, such that $$ E(c_0, \dots, c_n) \leq \|f\|_\infty + 1 $$ According to [Süli & Mayers] (p. 229) "The set $S$ is evidently bounded and closed in $\mathbb{R}^{n + 1}$". I see why $S$ is closed, but it is not evident to me why it is bounded. Any help will be appreciated.

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Works cited

[Süli & Mayers] Süli, Endre and Mayers, David F. An Introduction to Numerical Analysis. Cambridge University Press, 2003.

Answer 1

This is the consequence of three elements:

1. First you have $\Vert D(c_0,\dots,c_n) \Vert_\infty \le 2\Vert f\Vert_\infty +1$ for points belonging to $S$.
2. $(c_0,\dots,c_n) \mapsto \Vert D(c_0,\dots,c_n) \Vert_\infty$ is a norm of the finite dimensional space $\mathbb{R}^{n+1}$. And by point 1. $S$ is bounded for that norm.
3. Finally as in finite dimensional spaces all norms are equivalent, $S$ is bounded for whatever norm you choose.