= Solution
Let $E=\|f-p\|_\infty$. If $E=0$, minimality is immediate. Otherwise suppose a <polynomial> $q$ of degree at most $n$ has strictly smaller error. At each alternation point,
$$
(-1)^j(q-p)(x_j)=E-(-1)^j(f-q)(x_j)>0.
$$
Thus $q-p$ changes sign between every pair of consecutive $x_j$. By the <intermediate value theorem> it has at least $n+1$ distinct <roots of a polynomial>, contradicting its degree bound. Therefore
$$
\boxed{\|f-p\|_\infty\leq\|f-q\|_\infty\quad(\deg q\leq n).}
$$
This is the sufficiency direction of the <Chebyshev alternation theorem>.
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