= Solution
The <Weierstrass approximation theorem> states that the <polynomials> are a <dense subspace> of $C[0,1]$ in the <supremum norm>. By <linearity> and (c), $B_np\to p$ uniformly for every <polynomial> $p$. Given $f\in C[0,1]$ and $\varepsilon>0$, choose a <polynomial> $p$ with $\|f-p\|_\infty<\varepsilon$. The <operator norm> estimate in (a) gives
$$
\|B_nf-f\|_\infty\le\|B_n(f-p)\|_\infty+\|B_np-p\|_\infty+\|p-f\|_\infty\le2\varepsilon+\|B_np-p\|_\infty.
$$
First let $n\to\infty$, then $\varepsilon\downarrow0$. Hence \b[$\|B_nf-f\|_\infty\to0$ for every continuous $f$]. This is an instance of <extension of approximation operators from a dense subspace>.
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