This paper studies discrete-time invariant autonomous systems [GRAPHICS] We investigate the problem of when asymptotic stability of Sigma can be characterized by means of zero-state observability of Sigma and square summability of the output function h(x(k)). The relationship among observability, square summable series and stability of discrete-time autonomous systems is exposed. The main result developed in this paper unifies a series of stability theorems recently proposed in the literature, and generalizes Morse’s Lemma [Morse, A. S. (1990). Towards a unified theory of parameter adaptive control tunability. IEEE Trans. Autom. Contr., 35, 1002-1012.] in two directions : from continuous-time linear systems to their discrete counterparts and from linear systems to nonlinear systems.