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Invoking random coding, but not typical sequences, we give non-asymptotic achievability results for the major setups in multiuser information theory. No limitations, such as memorylessness or discreteness, on sources/channels are imposed. All the bounds given are powerful enough to yield the constructive side of the (asymptotic) capacity regions in the memoryless case. The approach relies on simple non-asymptotic counterparts of the packing and covering lemmas conventionally used in conjunction with the typical sequence approach.