= Solution
Fix $m$ and take $n\ge m$. Comparing the two products defining $f_{nm}$ and the <monomial> $g_m(x)=x^m$, change one factor at a time. Every factor has absolute value at most one on $[0,1]$, so the <triangle inequality> gives
$$
\|f_{nm}-g_m\|_\infty\le\sum_{r=0}^{m-1}\frac rn=\frac{m(m-1)}{2n},\qquad |f_{nm}(1)-1|\le\frac{m(m-1)}{2n}.
$$
The contraction proved in (a) and the <Bernstein falling-factorial identity> from (b) now give
$$
\begin{aligned}
\|B_ng_m-g_m\|_\infty&\le\|B_n(g_m-f_{nm})\|_\infty+\|(f_{nm}(1)-1)g_m\|_\infty\\
&\le\boxed{\frac{m(m-1)}n\longrightarrow0}.
\end{aligned}
$$
This proves <uniform convergence> for each fixed <monomial>; <linearity> then gives it for every fixed <polynomial>. The cases $m=0,1$ have zero error for every $n$.
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