Concerning your question about Hamlet's sanity in Shakespeare's Hamlet, I suggest a couple of different angles for you to consider.
First, the line of thought that you want to take is a bit difficult to prove (I'll get to that later), but the fact that Hamlet suffers from melancholy, or depression, is obvious to even a casual reader, and is well-accepted among critics: see his speech to Gertrude in Act 1.2.76-86, and his soliloquy in Act 1.2.129-159. It is easy to establish Hamlet's major depression. This should improve your ability to prove Hamlet's descent into actual madness from feigned madness.
Second, you may not want to attempt to deal in absolutes. In other words, you might want to take the angle that there are indications that Hamlet actually becomes mad, rather than taking the angle that he definitely becomes mad. I'm not a psychologist, so maybe I'm wrong and there is absolute proof. But this play is extremely ambiguous, and I don't know that there's absolute proof of anything when it comes to the question of why Hamlet waits, what he suffers from, etc.
That said, below is a quote from the enotes Study Guide on the themes in the play, one of which is madness and illusion and reality. The passage may give you ideas, but it may also serve as evidence for you, a second opinion.
As both a plot component and a central thematic cluster, madness and, with it, the line between reality and illusion are certainly prominent throughout Hamlet. The Prince feigns madness so well that we sometimes question his underlying sanity. Indeed, Hamlet himself harbors the fear that the ghost of his father may be an hallucination. Ophelia, of course, lapses into madness, sinking below the depths of a tragic tide of events into self-destructive melancholia. Reinforcing this, Shakespeare plays on the contrast between reality and illusion. This is most often brought out in contrasts between the "real" and the "seeming" kings of Denmark (see Hamlet's condemnation of Gertrude in Act III, scene iv). This notion of illusion is embodied in Ur-Hamlet's remarks about "my most seeming virtuous queen" (I.v) and in the "play-within-a-play" where mere illusion on the stage evokes the real emotion of guilt in Claudius, the "play" being the thing through which Hamlet "catches the conscience of the King" (II.ii).
For specific evidence, I suggest looking at the extreme behaviors Hamlet exhibits in his dealings with Ophelia and Gertrude, and his failure to show virtually any emotion over his causing the deaths of Polonius and his one-time friends Ros. and Guil.