= Solution
For any degree-at-most-$n$ <trigonometric polynomial> $t_n$, linearity and reproduction give
$$
f-v_{n,m}(f)
=(f-t_n)-v_{n,m}(f-t_n).
$$
The <operator norm> bound in the preceding part yields
$$
\|f-v_{n,m}(f)\|_\infty
\le\left(2+\frac{2n}{m}\right)\|f-t_n\|_\infty
\le2(M+1)\|f-t_n\|_\infty.
$$
Take the infimum over all such <trigonometric polynomials>. By the definition of <best uniform approximation>,
$$
\boxed{\|f-v_{n,m}(f)\|_\infty\le2(M+1)E_n(f)}.
$$
No choice of a minimizer is needed for this argument. It is an instance of the <polynomial reproduction error bound>: a bounded linear reproducing operator has error at most $1+\|P\|$ times the optimal error.
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