Once you begin an EdX course, the "View Course" button will change to "Resume Course". The "Resume Course" button remains as long as the course is still active and will not change to show the course as completed.
To see your progress toward completion you can view your course grading on the progress tab within your course. If the total column meets or exceeds the passing line then you have successfully passed the course. If you are in the free audit track you will not see any other notification of completion.
If you are registered in the verified track you will receive a certificate once you reach a passing score.You may share this certificate with others to show that you completed the course.