How did Paul prove his Roman citizenship?
Paul had dual citizenship between being Jewish and a Roman citizen. Because he was born in a Roman province, in the city of Tarsus, he automatically had Roman citizenship. This is just like American citizenship if you are born in America or whatever country you are born in.
Because he was Jewish, sometimes he allowed his Roman citizenship to be overlooked. But to get to Rome from Jerusalem, he allowed himself to be arrested and flogged, even though this is not permitted for Roman citizens (Acts 22-23).
Roman citizenship could be bought if you were a slave or did not grow up in a Roman province. One of the Roman citizens that flogged him in Jerusalem asked him about his Roman citizenship (Acts 22:26-28).
Being a Roman citizen allowed Paul to go the whole way to the Caesar in Rome, the Emperor, to defend his accusations. He also would be able to mention that they flogged him when you don’t flog a Roman citizen.
But Paul’s true mission to go to Rome was to spread the gospel of Jesus Christ. If he reached the Roman Emperor with the gospel, he would have achieved his goal. Paul spent much of his time speaking to the officials in each of the cities. He spread the gospel from the top down.
Paul used his dual citizenship wherever it helped him to spread the gospel the most. If it was to go to Rome in chains so that he could share the gospel with the most powerful person in the known world, he mentioned his Roman citizenship so he could present his case and the gospel before the Emperor.