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The encoding of sources with unknown statistics is considered. The average distortion that is obtained with universal coding schemes that are independent of the source statistics is shown to be asymptotically identical to the smallest average distortion that can be achieved with the best individual coding scheme (i.e., a code that is based on the specific statistics of the source). This result is shown to hold for any stationary source, as well as for a class of nonstationary sources. The discussion is limited to a certain important class of metric spaces.