A one-dimensional cellular automaton is a dynamical system which consisting in a juxtaposition of cells whose state changes over discrete time according to that of their neighbors. One of its simplest behaviors is nilpotency: all con gurations of cells are mapped after a nite time into a given \null" con guration. Our main result is that nilpotency is equivalent to the condition that all con gurations converge towards the null con guration for the Cantor topology, or, equivalently, that all cells of all con gurations asymptotically reach a given state.