= Solution
Apply the general <B-spline interpolation operator norm> estimate in part (I) to the inverse <norm> from (b):
$$
\boxed{\|P_{\mathbf x}\|_{L^\infty}\leq\|A_n^{-1}\|_{\ell^\infty}<3,}
$$
and hence in particular $\|P_{\mathbf x}\|_{L^\infty}\leq3$, independently of $n$. The strict finite-dimensional estimate uses the actual finite collocation <matrix>; it does not require periodic closure, natural boundary conditions or endpoint knot repetitions.
There is also a direct shorter verification of the uniform bound. For an index $i$ maximizing $|z_i|$, the <triangle inequality> gives
$$
\|A_nz\|_{\ell^\infty}\geq|(A_nz)_i|\geq\left(\frac23-\frac16-\frac16\right)\|z\|_{\ell^\infty}=\frac13\|z\|_{\ell^\infty},
$$
with an even larger margin when a neighbor is missing. This is the <inverse infinity-norm bound from diagonal dominance>, and again gives $\|A_n^{-1}\|_{\ell^\infty}\leq3$.
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