Old Cats - new puppy

We are looking at getting a new puppy this weekend, she is pretty small, part Shih tzu and part mini Dachsand. We have three old cats. One stays in the other bedroom and I am not worried about her but the other two are spoiled rotten and both are nearly 15. I keep reading that I need to introduce the three of them slowly. I am planning on crate training the puppy immediately and I have a play pen she can play in when we are home. Both cats are going to be quite a bit larger then the puppy. The cats will be able to leave the room anytime and go up to the attic if they want. But my elderly mom is worried that the cats will hurt the puppy and so she thinks we shouldn't get the dog. Anyone have answers on the best way to handle this? Thank you so much!

The puppy will be nine weeks old Friday.