= Solution
The <Korovkin theorem> says that a sequence of <positive linear operators on continuous functions> on $[0,1]$ converges uniformly to the identity on every <continuous function> if it does so on $1$, $x$ and $x^2$. Necessity is immediate; the quadratic barrier argument establishing sufficiency is given in Solution 6.
There is an endpoint misprint in the original PDF: the repeated knots at the right end must equal $1$, not $0$. Otherwise the advertised nondecreasing <spline knot sequence> on $[0,1]$ does not exist. Use the corrected clamped <spline knot sequences>, with fixed order $k$ and maximum gap $h_n\to0$. Let $m_n$ denote the number of <B-splines> for the $n$th sequence. Nonnegativity and partition of unity show that the <Schoenberg spline operator> $V_n$ is positive and $V_n1=1$.
The <monomial B-spline coefficients> reproduce the first two nonconstant test <polynomials>. Put
$$
a_{1,j}=\frac1{k-1}\sum_{r=j+1}^{j+k-1}t_r,\qquad
a_{2,j}=\binom{k-1}{2}^{-1}\sum_{j+1\le r<s\le j+k-1}t_rt_s.
$$
The pair sum uses $r<s$, as in the PDF; the converted TeX incorrectly includes $r=s$. Every sampling point $\tau_j$ and every knot entering these <coefficients> lie in $[t_j,t_{j+k}]$, whose length is at most $kh_n$. Thus $|\tau_j-a_{1,j}|\le kh_n$. Also all these numbers belong to $[0,1]$, so
$$
|\tau_j^2-t_rt_s|\le|\tau_j(\tau_j-t_r)|+|t_r(\tau_j-t_s)|\le2kh_n,
\qquad |\tau_j^2-a_{2,j}|\le2kh_n.
$$
The positive <B-spline> weights give
$$
\|V_nx-x\|_\infty\le kh_n,\qquad\|V_nx^2-x^2\|_\infty\le2kh_n.
$$
Together with $V_n1=1$, the three limits prove
$$
\boxed{\|V_ng-g\|_\infty\longrightarrow0\quad\text{for every }g\in C[0,1].}
$$
Under the usual interior knot multiplicities at most $k-1$, the <splines> are continuous and the quoted form of the <Korovkin theorem> applies directly. If full interior multiplicity is allowed, the output can be discontinuous. The same positive-operator quadratic barrier proof still works with bounded output <functions> and the <supremum norm>, and proves exactly the stated uniform error limit without asserting continuity of each output. Indeed the stronger <local-support error bound for a Schoenberg spline operator>, $\|V_ng-g\|_\infty\le\omega(g,kh_n)$, follows by comparing $g(x)$ with each active sample value. The restriction $k\ge3$ is needed for the displayed quadratic reproduction argument, not for this direct support estimate.
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