How to prove type soundness of Java-like languages without forgoing big-step semantics

Small-step operational semantics is the most commonly employed formalism for proving type soundness of statically typed programming languages, because of its ability to distinguish stuck from non-terminating computations, as opposed to big-step operational semantics. Despite this, big-step operational semantics is more abstract, and more useful for specifying interpreters. In previous work we have proposed a new proof technique to prove type soundness of a Java-like language expressed in terms of its big-step operational semantics. However the presented proof is rather involved, since it requires showing that the set of proof trees defining the semantic judgment forms a complete metric space when equipped with a specific distance function. In this paper we propose a more direct and abstract approach that exploits a standard and general compactness property of the metric space of values, that allows approximation of the coinductive big-step semantics in terms of the small-step one; in this way type soundness can be proved by standard mathematical induction.