In a persistent connection it is the connection that is persistent, not query results.
You're wanting a form of pagination. There are a couple of threads cited in the Coding Forum FAQ that go into this subject in great detail, and a couple of articles on this site. Your requirements would be quite a lot simpler than those threads, because what you're describing involes only one record per page; in the FAQ-linked threads, I suggest you look at the second one first.
Hint: you'll be learning how to use LIMIT in your SQL queries.