Take a look at how the border is applied. Sometimes people attach it with hot glue, staples, even wall paper paste -- we have seen it all! If everything is securely attached with wall paper paste the best option is to rent a steamer. This is going to loosen up and saturate the adhesive with steam making it easy for you to remove the wall paper in fairly large pieces. You will have to work in sections and really let the steam loosen everything up before you start peeling, don't use one of the paper cutting devices to shred it before steaming. That only means very small pieces to remove and could damage the drywall below. Once you have all the paper removed sand and repair any damaged area and be sure to prime the walls before you paint them.
Enjoy the fresh room!