Is there any evidence in the play that proves Hamlet's sanity?

After Hamlet has the meeting with his father's ghost in Act I, Scene 5, he swears Horatio and Marcellus to secrecy and tells them not to give any indication that they know anything of unusual import about him, regardless of

How strange or odd soe'er I bear myself—
As I perchance hereafter shall think meet
To put an antic disposition on—

This shows that he is already planning to act mad in the future. Why should he want to do that? Claudius is spying on him. Claudius is a very shrewd and dangerous adversary, and he is already suspicious. Hamlet feels it will be hard to act the same towards his uncle now that he knows all about him, and therefore he is thinking of putting on an act of insanity to keep Claudius from detecting what he is really thinking. He does succeed in convincing Claudius, Polonius, Ophelia, Rosencrantz and Guildenstern that he is mad.

In Act III, Scene 4, when Gertrude asks

What shall I do?

Hamlet tells her not to let "the bloat king"

Make you to ravel all this matter out,
That I essentially am not in madness,
But mad in craft.

Earlier in that scene he tells his mother

My pulse as yours doth temperately keep time(
And makes as healthful music. It is not madness
That I have utt'red. Bring me to the test,
And I the matter will reword; which madness
Would gambol from. 

Then in Act IV, Scene 2, he runs ahead of Rosencrantz and Guildenstern, pretending to be playing a game like Hide and Seek, which he calls

Hide fox, and all after.

Hamlet is obviously only pretending to be mad. We have seen that he is perfectly rational in his meeting with his mother. Then in Act IV, Scene 3, he continues to pretend that he thinks he is playing a child's game and that he thought Polonius was playing it too when he was hiding behind the arras. He tells Claudius that Polonius is at supper and goes into a long speech ending with

Your fat king and your lean beggar is but variable service -- two dishes, but to one table. That's the end.

Claudius at this point says

Alas, alas!

He believes his stepson is completely mad. But we know it is all a pretense. If Claudius did not think Hamlet was mad, he would probably be much more suspicious now and perhaps have Hamlet locked up. Hamlet foresaw such a possibility when he swore Horatio and Marcellus to secrecy in Act I, Scene 5. He needs his freedom if he is going to find a way to assassinate Claudius. It is essential that Claudius have no inkling that Hamlet has ever had any contact with his father's ghost. Claudius would be sure to think that Hamlet knows what nobody else in Denmark knows--that he killed his brother to usurp the throne and marry his wife.

So we see Hamlet pretending to be mad, and telling Horatio and Marcellus that he might do that, and also convincing his mother that he is not mad. On the other hand, we see no real evidence that he might be mad and not just pretending. It seems impossible that Hamlet could be mad sometimes and only pretending at other times.