= Solution
Choose a <best uniform approximation> $p_n\in\mathcal P_n$. It exists because this <vector subspace> is a <finite-dimensional vector space> and a minimizing sequence $q_j$ is bounded: $\|q_j\|_\infty\le\|f\|_\infty+\|f-q_j\|_\infty$. <Finite-dimensional equivalence of norms> then supplies a convergent subsequence of coefficients. If $E_n(f)>0$, the <Chebyshev alternation theorem> gives $n+2$ alternating error extrema. The <intermediate value theorem> supplies a zero $x_{n,k}$ of $f-p_n$ in each of the $n+1$ intervening open intervals. These <interpolation nodes> are distinct, and uniqueness of <polynomial interpolation> gives
$$
\boxed{\ell_n(f)=p_n,\qquad \|\ell_n(f)-f\|_\infty=E_n(f)\longrightarrow0}.
$$
The convergence follows from the <Weierstrass approximation theorem>. If $E_n(f)=0$, choose any $n+1$ distinct nodes: the <polynomial interpolant> is still $f=p_n$. This constructs <interpolation nodes from best uniform approximation>; the nodes depend on $f$, and no universal node system for all <continuous functions> is asserted.
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