A senior coworker is trying to get me fired, what can I do?

This coworker used to work in my position, and was told to train me when I arrived two years ago. Since then, although she has not been igned a supervisory role to me, she constantly goes through my work files and reports every little thing she doesn't like to my boss! Last week my boss came to me with an entire folder of photocopied "mistakes" that this woman had secretly collected, a few of which really were my errors but most of which were either completely insignificant or not mine at all. Needless to say this was shocking and infuriating. I kept my cool with my boss and promised to rectify all of the areas I had made any mistakes in, but this coworker has been undermining me with an extremely hostile single-mindedness all along, being rude in public and blaming me for anything that goes wrong even if it's not my error. She has been working with the company for 15 years and is good at her job so there is no way to go face to face with her and win. Am I doomed in this job?