They do not need proof because they already believe it. If they believe it, then it cannot be disproved.
Even if they believe something else that is exactly opposite of something else they believe, that does not make it false. Because the cynic believes both, then there is no argument because to the cynic, both are true.
There are two explanations for this kind of thinking. One, it is genetic and there is no hope of changing the person. The Greeks must have believed that because they executed their greatest cynic, Socrates, because his students brought Greece wrack and ruin.
The other is environmental, that the cynic has power, prestige, and possessions. With power, there is no argument. The powerful wins every time no matter what the argument. With prestige, the issue is privilege of place which wins every argument. This kind of cynic answers to no one beneath him/her. With possessions, the cynic buys whatever s/he wants to be true. Who can argue with success?
I suspect both explanations are true. But either way or both ways, philosophers had better start protesting the cynics of whatever stripe because cynics do not live in a truth-based universe. If they are in authority over us, they cannot help us or anyone else but themselves.