if someone doubts your word, how should you react

Move on, they're not your kind, never will be and will only believe what they want. Their own eyes witnessing you modeling what you told them may never be enough.

How do you know they disbelieved you if they offered no acknowledgment?
Remember, what we say is rarely the message received: each person interprets any message according to their own experiences. It's possible that something in their past experience (such as parents, siblings or others frequently lying) may have led them to distrust everyone. Or maybe they distrust particular queues. or maybe they were distracted or didn't hear.

First evaluate whether the relationship is important to you; if not, just walk away and forget it.
If it's important, just quietly and gently show them the proof.

I've had that happen a couple times. On here, I just let it go because, honestly, everybody here is doubtful and paranoid and I get that, totally.
IRL, I don't waste my time because I shouldn't have to prove myself nor do I need validation. I just keep on keeping on.

I wouldn't waste my time with them.