We make the case that the foundation for Rely-Guarantee reasoning can be fruitfully delivered by a coinductive semantics. Using insight from an Isabelle formalization, via a proof analysis we show that the coinductive semantics tends to simplify the proof development; in particular it enables more direct proofs for the soundness of the Rely-Guarantee rules. The comparison between inductive and coinductive proofs also suggests inductive counterparts of coinductive “up-to” enhancements. On the way, we fill a gap in the literature, by showing that three previously defined inductive semantics for Rely-Guarantee are equivalent. Underlying our transformation of an inductive into a coinductive semantics is the notion of inductively approximating a coinductive predicate—which, deployed in the opposite direction (from coinduction to induction), is a standard technical tool for approximating process algebra bisimilarities. On the spectrum between the abstract fixpoint theorems and concrete instances, we formalize effective format-based criteria that enable sound approximation.