Here's the general rule when it comes to articles:
Singular indefinite - A/an + noun
Singular definite - The + noun
Plural indefinite/indefinite and uncountable - (no article) noun
Plural definite/plural and uncountable - The + noun
I'm guessing that French is like Italian in that all plural nouns use an article, regardless of whether or not they're definite. In English, the distinction between definite and indefinite nouns is as important for plural or uncountable nouns as it is for singular nouns.
"Proof" is indefinite and uncountable here, so there's no article. If the speaker had already been looking for proof, and the listener knew, then you'd say "the proof" instead.