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The tracking differentiator was first proposed by Jingqing Han in 1989 and the proof of convergence was presented the first time in [J.Q. Han and W. Wang, Journal of Systems Science and Mathematical Science, 14(2)(1994), 177-183]. Unfortunately, the proof there is incorrect. This problem has been open for over two decades. In this paper, we give a rigorous proof under some additional conditions. An application to online estimate the unknown frequencies for the finite sum of the sinusoidal signals is presented. The numerical simulations illustrate the effectiveness of the estimation from both linear and nonlinear tracking differentiators.