In what way is The Tempest a play within a play?

The Tempest can be said to be a play within a play in that Prospero not only could be said to represent Shakespeare, but that the character of Prospero is quite deliberately directing all the action that is taking place on his island. He has set the scene for the action by causing the illusion of a shipwreck, just as a director of a play has to create the illusion of real action on stage. He directs all the players, from Ariel, to the Italian nobles, to his own daughter Miranda, practically feeding them their lines as a good prompter. He brings the characters together, and brings the show to a satisfying climax in true Shakespearean fashion.