Tbf, I don’t think Jim convinced her. They weren’t dating at the time, iirc.
she followed to be with him.
There was a weird gap from what I remember. They weren't together when they showed them in Stamford, but the first episode where Jim was back in Scranton, he was dating Karen even though Pam was single when he came back. The show didn't really do a good job filling in that gap of how Jim and Karen got together.