New house. One of the doors is too low.

So we moved into a new house and discovered that the door leading to the basement is hung just a little too low so that we can't put anything like a welcome mat or rug next to that door. It opens fine if the floor is bare but we really want to put a mat next to the other door that is next to this one. Everything around the door is finished so re framing is not really an option.

I talked to a local hardware guy and he said I could "shave the bottom of the door". Ideas?