Infected thumb?

The other day I had the annoying piece of skin on the side of my thumb nail and it was peeling so I just took a pair of small old scissors and clipped it and didn’t think anything of it as I’ve done it before in the past.

Well, I’m pretty sure it’s infected, as it’s been really sore today whenever I touch it.

I don’t see puss or anything. But I did put some peroxide- nothing fizzed and some neosporin. But yeah I’m starting to worry that I have an infection and bottom pad of my thumb is starting to bother me too