Which one makes more sense to you? A or B

Proving

You need to follow the prepositional phrase closest to with a word (or construction) that functions as a noun. (Here, the word to is not part of an infinitive, rather a preposition.)

The word prove forms part of the infinitive verb form to prove, whereas the word proving is the present participle form of the verb to prove functioning as a noun – this is called a gerund.

Oswald Avery, a Canadian bacteriologist, came closest to proving that DNA is a hereditary molecule.