  We study the convergence aspects of the metric on spectral truncations of
geometry. We find general conditions on sequences of operator system spectral
triples that allows one to prove a result on Gromov-Hausdorff convergence of
the corresponding state spaces when equipped with Connes' distance formula. We
exemplify this result for spectral truncations of the circle, Fourier series on
the circle with a finite number of Fourier modes, and matrix algebras that
converge to the sphere.
