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In this paper, we propose a Dickey-Fuller difference statistic to sequentially detect the change-point that shift from an unit root process to a long-memory process. The limiting distribution of monitoring statistic under the unit root process null hypothesis as well as its consistency under the alternative hypothesis are proved. Simulations indicate that the new method can control the empirical size well even for the heavy-tailed unit root process when using the sieve bootstrap method computing its critical values. In particular, it performs significantly better than the available method in the literature under the alternative hypothesis. Finally, we illustrate the new monitoring procedure by a set of foreign exchange rate data.