Catherine is right, the bugs stay on the head (luckily) and you will not find them all over the home. Also, IF the nits fall off they cannot hatch without being in their handy hair incubator.
I would recommended that you check everyone else in the house just to make sure. If the comb you purchased is on of the nice metal ones you can use it to do head checks. Just wet the hair, run the metal comb through and wipe it on a white paper towel. This way you can see if there are any nits or bugs in the persons hair.
Wet combing with a good metal comb will eventually get rid of them all, it just takes time and patience. Focus your energy on getting all the nits out and enjoy your now clean home.
If you have any other questions let me know, I happen to treat head lice for a living, I know they are maddening, but it sounds like you are doing a great job :)
Nit Nanny