You have been given some good advise about the flea bombs but the problem started because you let the cat go outside. Cats need to live inside and if you can’t handle a litterbox........ then I think the cat is screwed.
Be careful with the flea treatment on the cat, we killed a can because we put too much on him. Also you can spray your yard to kill fleas before they get into you house. You can bring fleas in on your own legs and feet, they need to be eliminated out there too.
You have been given some good advise about the flea bombs but the problem started because you let the cat go outside.
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In a way that’s correct, but I know cats that stay outside all the time and don’t have fleas. They don’t get any kind of treatment either. The problem is once the fleas get inside they breed like crazy in the carpet and upholstery.