The problem of convergent fault estimation for a class of linear systems with faults and disturbances is addressed for the first time in this technical note. A differential-equation-based convergence theorem of mean sequence of estimates is proposed, which provides an iterative mean estimation technique that has not been fully reported yet in the literature. Based on the technique, a convergent fault estimation strategy for the linear system in consideration is designed, and the convergence of mean sequence of fault estimates is proved theoretically. An iterative algorithm is given subsequently to compute the fault estimates. Numerical simulation results demonstrate the effectiveness and merits of the proposed method.