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This paper sheds light on adaptive coding with respect to classes of memoryless sources over a countable alphabet defined by an envelope function with finite and non-decreasing hazard rate (log-concave envelope distributions). We prove that the auto-censuring (AC) code is adaptive with respect to the collection of such classes. The analysis builds on the tight characterization of universal redundancy rate in terms of metric entropy and on a careful analysis of the performance of the AC-coding algorithm. The latter relies on nonasymptotic bounds for maxima of samples from discrete distributions with finite and nondecreasing hazard rate.