Let $f\in C([a,b])$, and let $(\tau_m)$ be a sequence of knot partitions of $[a,b]$ whose mesh size tends to $0$. For each fixed degree $r\ge 0$, there exist splines $s_m$ of degree at most $r$ subordinate to $\tau_m$ such that \begin{align*} \|f-s_m\|_\infty \to 0. \end{align*} More precisely, if $1\le k\le r$ and $f\in C^k([a,b])$, then the splines may be chosen so that \begin{align*} \|f-s_m\|_\infty \le C_{r,k}\,h_m^k\,\omega(f^{(k)},h_m), \end{align*} where $h_m$ is the mesh size of $\tau_m$, $\omega(g,h)=\sup\{|g(x)-g(y)|:|x-y|\le h\}$ is the modulus of continuity, and $C_{r,k}$ depends only on $r$ and $k$. If $f\in C^{r+1}([a,b])$, then a corresponding endpoint estimate has the form \begin{align*} \|f-s_m\|_\infty \le C_r h_m^{r+1}\|f^{(r+1)}\|_\infty. \end{align*}