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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 $mu>1/(m-1)$, 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 a functional analytic foundation and an extension to arbitrary Lewis numbers to the numerical studies of Sherratt & Marchant (IMA J. Appl. Math. extbf{56}, 1996, pp. 289--302) and the asymptotic analysis of Needham & Barnes (Nonlinearity extbf{12}, 1999, pp. 41--58).