We provide a construction that, given a big-step semantics describing finite computations and their observations, extends it to include infinite computations as well. The basic idea is that the finite behavior uniquely determines the infinite behavior once observations and their composition operators are fixed. Technically, the construction relies on the framework of inference systems with corules. The effectiveness and scope of the approach are illustrated by several examples. The correctness is formally justified by proving that, starting from a big-step semantics equivalent to a reference small-step semantics, this equivalence is preserved by the construction.