The semantics of "faith"
I have problems with both parts of this statement. Seems to me that when our theological texts refer to "faith", or being "faithful" (ne'eman), they really mean faith in the sense of steadfastness, fidelity. In other words, not abandoning a belief in the face of persecution or dissuasion, like a good friend who sticks by you through thick and thin. I don't believe it means to acquire a belief without a rational basis for it. So to appeal to the usage of this word seems disingenuous, or at least inappropriate in this context.
Tackling the first part of the statement - it is probably true that you could not prove the existence of G-d, just like you could not prove the existence of the Flying Spaghetti Monster. However, this is not what the crux of the argument is usually about. The proof that is sought is usually the veracity of claims of various religions about G-d's interaction with the physical universe and specifically direct communication with various individuals. This is something that should be easy to prove.