= Solution
Write each <polynomial> of <degree of a polynomial>[degree] at most $n$ as $T\mathbf a$ and define
$$
F(\mathbf a)=\lVert T\mathbf a-f\rVert_\infty.
$$
This is a <continuous function>, and the reverse <triangle inequality> together with part (a) gives
$$
F(\mathbf a)\geq \lVert T\mathbf a\rVert_\infty-\lVert f\rVert_\infty
\geq\delta\lVert\mathbf a\rVert_2-\lVert f\rVert_\infty.
$$
Consequently $F$ is a <coercive function>. Choose $R$ so large that $F(\mathbf a)>F(\mathbf0)$ outside the closed <Euclidean ball> of radius $R$. That ball is <compact>, so the <extreme value theorem> supplies a minimizer $\mathbf a_*$ there. Then $P=T\mathbf a_*$ is a <best uniform approximation> to $f$ among polynomials of degree at most $n$.
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