d) What proof does Polonius provide Gertrude with to show that Hamlet is what he refers to as "mad"? What is Gertrude's response to Polonius?

In act 2, scene 2, Polonius, in his characteristically long-winded way, tells Gertrude that her son is mad. Brevity may be the soul of wit, but, as usual, there is nothing particularly brief about Polonius' explanation. However, he gets there eventually:

I will be brief: your noble son is mad.
Mad call I it, for, to define true madness,
What is ’t but to be nothing else but mad?

Polonius proceeds to make his case on the basis of a love letter written by Hamlet to Ophelia. Hamlet has been assiduously courting Ophelia for some time. However, Polonius has told his daughter that Hamlet, as a prince, is out of her league and that she should repel his advances. Hamlet's response to Ophelia's rebuff is to go mad with grief (or so it appears, at any rate). He is so "mad" with unrequited love he will not even touch his food.
 
Gertrude suspects that all is not well with Hamlet. She has seen him walking up and down the lobby for hours at a time. She does not want to believe her son is mad, but neither she nor Claudius have ever had any reason to doubt Polonius's probity. When she sees Hamlet enter, reading a book and looking incredibly sad, she starts to fear the worst.