A bound is derived for the norm of the difference between the solution of a linear differential equation with known input, and an approximate solution obtained by expansion in a Chebyshev series. The approach is to derive a bound on the difference between two approximate solutions of different order. This bound is obtained in terms of the order of the approximation of lowest degree, the difference in the orders of approximation, and certain matrix norms. Extending this expression so that it is valid for arbitrarily large differences in order yields a norm bound on the difference between the approximate solution and the exact solution. The basic approach can be extended to other orthogonal function sets.