Is this a bad infection?

A couple weeks ago, I got a minor cut at work while working with files. I didn't think much of it at first thinking it would heal like normal.

At first, I just noticed the pain was not going away. Every time I touched it wrong it would be painful. Again I didn't think much of it.

However, a few days ago, I began to notice swelling in the area above the cut. This was the point I started using triple antibiotic ointment, as I thought it was better late than never. However, it seems like it has only gotten worse since and every time I hit it wrong, the pain is nearly unbearable for a few seconds. I also notice a white spot around my fingernail that has since become rather green.

I am not one to usually go to doctors unless it is absolutely necessary. However, given that this infection doesn't seem to be getting any better, I have been wondering if I may need to go, or if this should eventually die down.

For background, I am 23 and have no major medical issues that I am aware. I do not smoke and hardly ever drink. I take no medication outside of a regular multivitamin.

Any advice is greatly appreciated.