= Solution
The <Korovkin theorem> states that if <positive linear operators on continuous functions> $L_n:C[0,1]\to C[0,1]$ satisfy
$$
\|L_ne_j-e_j\|_\infty\longrightarrow0,\qquad e_j(t)=t^j,\quad j=0,1,2,
$$
then $\|L_nf-f\|_\infty\to0$ for every real <continuous function> $f$.
Let $r=k-1\ge2$ and $h_n=|\Delta_n|$. For the <B-splines> in this question, $N_{i,n}\ge0$ and their sum is one on the basic interval. Thus the <Schoenberg spline operator> $V_n$ is linear and positive. To verify the three test <functions>, suppress the index $n$ and write the <Marsden identity> with $\omega_i(x)=\prod_{j=1}^r(x-t_{i+j})$. Comparing the coefficients of $x^r,x^{r-1},x^{r-2}$ gives the <monomial B-spline coefficients>
$$
1=\sum_iN_i(t),\qquad
t=\sum_i\xi_iN_i(t),\qquad
t^2=\sum_iq_iN_i(t),
$$
where
$$
\xi_i=\frac1r\sum_{j=1}^rt_{i+j},\qquad
q_i=\frac1{\binom r2}\sum_{1\le j<\ell\le r}t_{i+j}t_{i+\ell}.
$$
The first coefficient is the partition of unity; the second is the <Greville abscissa>. Every internal knot and the sample point $\tau_i$ lie in $[t_i,t_{i+k}]$, of width at most $kh_n$. Consequently $|\tau_i-\xi_i|\le kh_n$. Since all these numbers lie in $[0,1]$, for any two internal knots $u,v$ we also have
$$
|\tau_i^2-uv|\le|\tau_i|\,|\tau_i-u|+|u|\,|\tau_i-v|\le2kh_n.
$$
Averaging gives $|\tau_i^2-q_i|\le2kh_n$. Nonnegative weights of sum one now yield
$$
\|V_n1-1\|_\infty=0,\qquad
\|V_ne_1-e_1\|_\infty\le kh_n,\qquad
\|V_ne_2-e_2\|_\infty\le2kh_n.
$$
Here the <spline> order $k$ is fixed, so all three errors tend to zero. The <Korovkin theorem> therefore proves
$$
\boxed{\|V_nf-f\|_\infty\longrightarrow0\quad\text{for every }f\in C[0,1].}
$$
No special selection such as the <Greville abscissae> is required for the sample points. In fact the <local-support error bound for a Schoenberg spline operator> also gives $\|V_nf-f\|_\infty\le\omega(f,kh_n)$. Under the usual continuous-spline convention the operators have the codomain required by the stated theorem. If the permissive knot notation is taken to allow full interior multiplicities and hence discontinuous <splines>, the same conclusion in the uniform <norm> still follows from this direct bound, although each approximant need not then be continuous.
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