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Successive interference cancellation (SIC) is a technique for increasing the capacity of cellular code-division multiple-access (CDMA) systems. To be successful, SIC systems require a specific distribution of the users' received powers, especially in the inevitable event of imperfect interference cancellation. This apparent complication of standard CDMA power control has been frequently cited as a major drawback of SIC. In this paper, it is shown that surprisingly, these "complications" come with no additional complexity. It is shown that 1-bit UP/DOWN power control-like that used in commercial systems-monotonically converges to the optimal power distribution for SIC with cancellation error. The convergence is proven to within a discrete step-size in both signal-to-noise plus interference ratio and power. Additionally, the algorithm is applicable to multipath and fading channels and can overcome channel estimation error with a standard outer power control loop.