Other mistake: Harry commandeers one of the AV8-A's to go to save his daughter, and makes a clumsy vertical takeoff (where he bumps the police cars, etc). The reality is that, besides the deafening noise, the AV8-A generates a strong down wash when it takes off vertically which would have blown all the people on the bridge (and even some of the cars) in all directions. A few seconds later there is a cut to a real AV8-A vertical takeoff over water where we can see the effect of a real down wash on the water's surface. (02:02:20)
Other mistake: In a scene where Gib and Harry are getting into their car, if you look very closely you can see a elderly old lady staring into the camera.
Other mistake: During the bathroom shootout scene, the terrorist shoots the bathroom stall walls from a heavy angle. When the walls and doors are shown from the front while being opened, the bullet holes go directly through both walls of the doors, which would only look like that if they were shot from directly in front.
Other mistake: When Harry fires the Harrier's Sidewinder missile with the lead terrorist attached to it, he has to switch on the "master arm switch". That switch would have already have been enabled in order to fire the Harrier's gun.
Chosen answer: He didn't really. Harry just knew he was heading up. He pressed the top floor to get the elevator moving and kept an eye on him on the way up. If the terrorist's elevator suddenly stops, Harry can press the button for that floor to stop his elevator.
XIII