- Semiconstrained systems were recently suggested as a generalization of constrained systems, commonly used in communication and data-storage applications that require certain offending subsequences be avoided. In an attempt to apply techniques from constrained systems, we study sequences of constrained systems that are contained in, or contain, a given semiconstrained system, while approaching its capacity. In the former case we describe two such sequences resulting in constant-to-constant bit-rate block encoders and finite-state encoders. Perhaps surprisingly, we show in the latter case, under commonly- made assumptions, that the only constrained system that contains a given semiconstrained system is the entire space. A refinement to this result is also provided, in which semiconstraints and zero constraints are mixed together.