How to handle troublesome coworker?

I have to work relatively closely with a coworker who constantly undermines me. I'm not sure if he has difficulties with social skills/is unaware how he comes across or if he's intentionally manipulative. When I've confronted him about it, he says he doesn't mean anything that way, but his condescending, meddling behavior doesn't change. If I forget something, but catch myself/make an adjustment on my own, he'll try to remind me of my mistake and take credit for how I fixed it. Or he'll over-exaggerate how bad something I prepared resulted (he did the diagnostic); when I look at the results the next day, they're borderline-OK and when I also repeat the diagnostic, the results are great. He has a history of undermining other people/making himself look better by questioning other other people's work (even our supervisor), and has been reprimanded for *his* job performance. I recently received a promotion (he is much older, but I have more applicable experience/work more efficiently).