I have introduced a puppies to an older cat, the second time was probably worse than the first, but they lived together happily in the end.
Initially the cat would hiss even from the other side of the patio door, and we had to keep them apart as I was frightened of the puppies eyes. But as puppies sleep a lot, I started to let the cat come in and 'inspect' the puppy while he was asleep, obviously under observation from me in case puppy woke up. Then we started to allow them together with the puppy on a thin slip lead so we had control and he couldnt chase the cat, everytime he 'lunged' forward to the cat, we said No, and went from there. It took a few months to be able to trust them completely, but they all lived happily ever after not taking any notice of each other.......although if the dogs saw another cat, that was another story.