Problem: With Apple mail, sometimes messages will be marked as junk, yet stay in the inbox. After being deleted, they will reappear in the inbox.

Solution: First, be sure to move (save) any messages from the inbox that you want to keep. Then, it is necessary to go to user folder > library > mail > account, messages, and then find the inbox for the selected account. Assuming the inbox is empty except for the messages that need to be deleted, it is possible to remove the contents of the inbox. Remove the files you find in that folder.