We consider a reaction-diffusion system modelling propagating fronts of an autocatalytic reaction of order m in a one-dimensional, infinitely extended medium. The Lewis number, i.e. the ratio of the molecular diffusivity of the autocatalyst to that of the reactant, is arbitrary. We prove that if the initial profile of the front decays exponentially or algebraically with exponent greater than 1/(m-1), the speed of the front is bounded for all times. Our method relies on weighted Lebesgue and Sobolev-space estimates, from which we can reconstruct pointwise results for the decay of the front via interpolation. The result gives both, a functional analytic foundation, and an extension to arbitrary Lewis numbers, to the numerical studies of Sherratt and Marchant (IMA J. Appl. Math. 56, 1996, pp. 289-302) and the asymptotic analysis of Needham and Barnes (Nonlinearity 12, 1999, pp. 41-58).