The notion of approximate truth has never been clearly and generally defined, but the intuitive idea is clear enough. There are many qualities we attribute to a class of objects, and the set of qualities that we truly attribute to those objects increases over time. This notion means different things to different people, but is generally taken to refer to the increasing ability science gives us to manipulate the world, predict natural phenomena, and build more sophisticated technology. Convergent realists often argue for their position by pointing to the increasing success of science. Scientific theories are empirically successful.
Scientific theories are approximately true.
Scientific theories are empirically successful. The terms in scientific theories refer to real objects. These are needed to make the argument even prima facie plausible. That said, let's go with the stronger argument that uses the conditionals stated above. Without the premises in brackets, these arguments are based on inference to the best explanation of the success of science.