= Solution
{c}
For $0\leq t\leq1$, the <triangle inequality> gives
$$
|p_n(t)-p(t)|\leq\sum_k|a_{n,k}-a_k|t^k\leq\sum_k|a_{n,k}-a_k|=\rho(p_n,p).
$$
Taking the <supremum> proves $\boxed{d(p_n,p)\leq\rho(p_n,p)}$. Hence convergence for $\rho$ implies convergence for $d$.
The converse is false. For $n\geq1$, take $q_n(t)=(1-t)^n/n$ and $p=0$. Its <supremum norm> is $d(q_n,0)=1/n\to0$, whereas the <binomial theorem> gives
$$
\rho(q_n,0)=\frac1n\sum_{k=0}^n\binom nk=\frac{2^n}{n}\longrightarrow\infty.
$$
The alternating coefficients cancel on $[0,1]$; the <polynomial coefficient sum norm> measures their <absolute values> before any cancellation.
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