Why did Paul keep talking about his sufferings when defending his apostleship?

Les teaches that as Paul went among Gentile cities calling out a people for Christ, Jews, whether believing or still in Judaism, constantly attacked him, saying he never walked with Jesus like Peter and was just an upstart bringing this on his own. Under those attacks, Paul had to defend his apostleship, and over and over he referred to the fact that he had suffered and suffered so Gentiles could get the message of grace. Les explains this is basically the theme of the first six chapters: Paul again defends his apostleship and shows the Corinthians all he had to go through for the sake of the gospel. The Judaizers, who believed Jesus was the Christ and were saved under the kingdom gospel but did not comprehend God's grace and that the law was satisfied and set aside, followed Paul and told his converts they could not be saved on Paul's gospel alone; they still had to practice Judaism, be circumcised, and keep the law. Les notes this just about drove the apostle up the wall, and he suffered because of that.