"How can faith be proved?"

"One cannot prove anything here, but it is possible to be convinced."

"How? By what?"

"By the experience of active love. Try to love your neighbors actively and tirelessly. The more you succeed in loving, the more you'll be convinced of the existence of God and the immortality of your soul. And if you reach complete selflessness in the love of your neighbor, then undoubtedly you will believe, and no doubt will even be able to enter your soul. This has been tested. It is certain...
Active love is a harsh and fearful thing compared with love in dreams. Love in dreams thirsts for immediate action, quickly performed, and with everyone watching. Indeed, it will go as far as the giving even of one's life, provided it does not take long but is soon over, as on stage, and everyone is looking on and praising. Whereas active love is labor and perseverance, and for some people, perhaps, a whole science...in that very moment when you see with horror that despite all your efforts, you not only have not come nearer your goal but seem to have gotten farther from it, at that very moment...you will suddenly reach your goal and will clearly behold over you the wonder-working power of the Lord, who all the while has been loving you, and all the while has been mysteriously guiding you."
--The Brothers Karamazov, Book 2, ch. 4.