= Solution
For each $n$, choose the <best uniform approximation> $p_n^*$. If its error is nonzero, take its $n+2$ alternating extrema. The <intermediate value theorem> supplies a point $y_j$ strictly between the $j$th and $(j+1)$st extrema such that $f(y_j)=p_n^*(y_j)$, for $0\le j\le n$. These $n+1$ points are distinct. The <polynomial interpolation> problem at them has a unique solution of degree at most $n$, so its solution is exactly $p_n^*$. This constructs the <interpolation nodes from best uniform approximation>.
If $E_n(f)=0$, then $f$ itself belongs to the <polynomial> space, and any $n+1$ distinct nodes give the same conclusion. In either case, take $x_{n,j}=y_j$ or the arbitrary nodes just described. Then
$$
\|\ell_n(f)-f\|_\infty=E_n(f).
$$
The <Weierstrass approximation theorem> gives $E_n(f)\to0$; alternatively, the bound derived in part 4(b), combined with <uniform continuity>, gives this convergence directly. Therefore
$$
\boxed{\|\ell_n(f)-f\|_\infty\longrightarrow0.}
$$
The nodes may depend on $f$ and on $n$. The conclusion does not assert a universal sequence of interpolation grids working for every continuous target.
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