Hi, I had a flea problem a few months back and learned a few things.
First a flea spends about 10% of its time on your cat. the rest of the time is on your carpet, bedding, etc. Since your baby is so young I strongly recomend not using a topical flea treatment like advantage. What you can do is treat it completely naturaly. I refuse to use any pesticide treatment what so ever on my fur baby. Here is what I did to win what I call World War Flea.
First cover your floor in Borax, a laundry detergent that is non toxic to pets and kids. When I say cover I mean cover, we live in about 700 square feet and it took 3 boxes. Once you put it on your floor sweep it in with a broom so it gets down to the base of the carpet. Wait aboput 2 days before you vacume. It sufficates the fleas and kills them. Vibration is what couses flea eggs to hatch, so throw out the vacuum bag after you vsacuum. While you are waiting for it to do its thing bath your little one in Dawn dish washing liquid. It kills fleas on conact. After that wash all your bedding in hot water. That will kill the fleas there. The borax in the floors will last for about a year. It will take about a week for you to really notice a major diferance but it will work. I have been flea free for almost 6 months now and have not had to introduce any type of chemical into my home.