2001 Hyundai Sonata Battery/Alternator

Car stopped and made clicking sounds when I tried to re-start. It started in a few min but stopped again. AutoZone said bad cell in battery, purchased new battery, still had battery light on inside car with new battery, checked alternator, AutoZone said alternator bad, drove car to and from work on new battery w/out working alternator, installed new alternator, battery light still on because it had been run the previous day w/out a working alternator. My husband thinks something else is wrong and thinks the alternator should charge the battery even when it's this low. I think that we should have the battery charged back up and not rely on the new alternator to do it. Please, I need your help!