= Solution
Take any <trigonometric polynomial> $t\in\mathcal T_n$. The reproduction property in (a) gives
$$
f-v_{n,m}f=(f-t)-v_{n,m}(f-t).
$$
By (b),
$$
\|f-v_{n,m}f\|_\infty\leq\left(2+\frac{2n}{m}\right)\|f-t\|_\infty.
$$
Taking the infimum over $t\in\mathcal T_n$ yields the <polynomial reproduction error bound> for this operator. With $E_n(f)=\inf_{t\in\mathcal T_n}\|f-t\|_\infty$ and $n/m\leq M$, it becomes
$$
\boxed{\|f-v_{n,m}f\|_\infty\leq2(M+1)E_n(f).}
$$
This is a near-best <uniform approximation> estimate; it follows from reproduction and boundedness without requiring the <de la Vallée Poussin sum> itself to be a best approximant.
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