  Establishing the convergence of splines can be cast as a variational problem
which is amenable to a $\Gamma$-convergence approach. We consider the case in
which the regularization coefficient scales with the number of observations,
$n$, as $\lambda_n=n^{-p}$. Using standard theorems from the
$\Gamma$-convergence literature, we prove that the general spline model is
consistent in that estimators converge in a sense slightly weaker than weak
convergence in probability for $p\leq \frac{1}{2}$. Without further assumptions
we show this rate is sharp. This differs from rates for strong convergence
using Hilbert scales where one can often choose $p>\frac{1}{2}$.
