Borrowed cat has fleas and has brought them in the house

[BercowsFestiveFlamingo]

How do I get rid of them?
We love borrowed cat and don't want to ban him from the house. I've told his owner he has fleas and she said she'd treat him. I've flea combed him a few times and removed a fair few each time but I've found a flea on my arm today and the dc have had to odd one on them.

Get a desk lamp and put it on the floor over the top of a plate of soapy water - leave it on at night and any fleas will be attracted to the light, fall into the water and drown.

If you don't use soap (it only needs a drop of liquid detergent) they can get out.

Don't be tempted to try to administer any treatment to the cat as there is a possibility they may be allergic to it.

Thank you. I'd never give him anything. We don't feed him, he just likes all the fuss he gets from us.
I've done some thorough vacuuming today so that should help