= Solution
For a real <continuous function> on $[-1,1]$, the <Chebyshev alternation theorem> gives a unique <best uniform approximation> $p_n^*$ from the space of <polynomials> of degree at most $n$. If $f$ is not already in that space, a candidate $p$ is best if and only if there exist ordered points $x_0<\cdots<x_{n+1}$ and a sign $\sigma\in\{1,-1\}$ with
$$
f(x_j)-p(x_j)=\sigma(-1)^j\|f-p\|_\infty,
\qquad 0\le j\le n+1.
$$
Thus a nonzero best error has \b[at least $n+2$ alternating extrema of equal magnitude]. If $f$ is already a <polynomial> of degree at most $n$, the unique best approximant is $f$ itself with zero error.
Existence does not require assuming that the infimum is attained: a minimizing sequence has uniformly bounded <polynomial> <norms>, and evaluation at $n+1$ fixed distinct nodes bounds its coefficients through an invertible interpolation <matrix>. A convergent coefficient subsequence then attains the infimum. The alternation criterion characterizes that minimum and gives uniqueness. In the following parts, $n\ge1$ whenever $E_{n-1}$ occurs.
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