How can you prove Hamlet's sanity?

Hamlet is a great tragedy by William Shakespeare. Here we find that Hamlet behaves like a mad man. In the play, we find that Hamlet is very much shocked to have been informed from the ghost of his father that his father is killed by his uncle. 

Hamlet senses some fowl and feels insecure. He wants to confirm the statement of the ghost. This is why, he decides to put a feigned madness. But his madness or sanity can be proved in several ways. Now let us try to prove whether Hamlet is mad or not. His madness is feigned. He acts very normally in this play. He talks rationally. He shows great intellectual power in his conversation with Horatio. 

His madness or sanity is proved when he enacts the play within-play, The Murder of Gonza:o. His soliloquies also prove his madness or sanity very clearly. His soliloquies are logical and full of wisdom and deep thinking. His reasoning shows him as a philosopher and as a scholar. The words, ideas and feelings used in his soliloquies cannot be regarded as a madman. 

These facts prove that Hamlet’s madness is feigned. He pretends to be mad but he is actually not so. Certainly, he must have causes. He does it with a view to taking revenge upon his enemies and saving himself any impending danger.