Near the end where Hal is trying to get Catherine to stay, they have a face to face conversation in the street and Catherine's hair is pulled front and back and front again in different shots.
When Catherine and her father take dinner and they are talking about how many days she missed by being lazy, her hair position alternates from a shot to another. Her fringe is first on the sides of her face, then in the next shot it's pulled back to her ponytail, then back again on the sides in the next shot.
When Catherine and Hal are talking on the front porch the morning after the funeral reception, the labels on the beer bottles alternate between facing the camera and facing away from the camera.
Items on the floor change position after Catherine empties the backpack.
Catherine's arm changes position when Claire talks about vegetarian chili.
When Dobbs comes in on the sisters and gushes about the Proof, he says it may prove a mathematical theorem, but a theorem is something already proven, such as Pythagoras' Theorem, and is already taught as fact. What he may meant to have said that it might prove a theory.
Near the end when Catherine says she'd rather go to sleep than talk through her father's proof, his mouth keeps moving after he stops yelling.
In the scene where Catherine is talking to her father by a park bench about going to school, the backdrop of the city and the sun is moving slowly from left to right
When Catherine and Claire are being driven to O'Hare Airport to fly to NYC, the car is shown heading east toward Downtown Chicago away from the airport rather then west toward the airport.
The daughter talks about how the father was looking for a message from aliens in the Dewey decimal call numbers on the books from the University of Chicago library. The University of Chicago uses Library of Congress call numbers, which begin with letters, not numbers.