Solution
= Solution
The <Schoenberg spline operator> is positive and $V_n1=1$. If $\xi_i=a_{1,i}$, then both $\tau_i$ and $\xi_i$ lie in $[t_i,t_{i+k}]$, so
$$
\lVert V_nt-t\rVert_\infty\leq k|\Delta_n|\to0.
$$
Because $a_{2,i}$ averages products of knots in the same interval and all points lie in $[0,1]$,
$$
\lVert V_nt^2-t^2\rVert_\infty\leq2k|\Delta_n|\to0.
$$
The <Korovkin theorem> now proves
$$
\boxed{\lVert V_n(f)-f\rVert_{C[0,1]}\to0}
$$
for every $f\in C[0,1]$.