Continuity mistake: While Lois and Superman are being mugged in the alley, Lois kicks the thief and protects herself leaning with her face against the wall and her arms widely spread. A frame later, when Clark grabs the bullet, her arms are close to her head.
Revealing mistake: As Air Force One enters the frame from the right, there appears to be a set of tracks faintly visible under it.
Revealing mistake: When the helicopter is hanging off the roof, a safety securing cable is attached to the fuselage to keep it in place.
Continuity mistake: When Superman lifts the plane during the storm, after the pilot asks what's going on, the copilot glances at the window grabbing its side. From the opposite angle he is away, repeating all movements.
Revealing mistake: When Superman and Lois begin their flight from her apartment, a pair of wires suspending each of them are visible, especially on the Blu-ray.
Revealing mistake: All the footage of the missiles being launched is of a poor quality and grainy, which is inconsistent with the good quality of the rest of the shots, revealing that it's stock footage.
Continuity mistake: In the extended edition, before Superman enters Lex's lair his hair is short, then when he's being shot at, his hair is much longer and bulkier. When he finally enters the place, it's back to short.
Revealing mistake: Superman has just saved Jimmy at the dam and is holding him in the air. Jimmy's fringe is standing up thanks to some hair fixer to achieve the effect of wind blowing (though it's a green screen effect). When Superman sets him on the floor his hair remains up giving away the trick, until the angle changes and it looks all combed and nice.
Continuity mistake: The two cops are following Otis at Grand Central Station and two girls, one in yellow, pass by their left side. When the angle changes to a closer one, the girls are in the back and pass by again.
Continuity mistake: When Clark shows Perry a white paper with directions of where to send the payroll, the way he holds the paper differs between shots.
Continuity mistake: Lois exits Perry's office and Clark enters to talk about salary. His hat swaps from facing up to down between shots.
Continuity mistake: When Clark is about to jump in front of the train, there's a crossing sign shaped as an X. Depending on the shot the front upper left part is either bent and about to break, straight, or missing.
Continuity mistake: When the Kent's stop the truck and see the cradle, the sky swaps from sunny and clear to heavily cloudy in a second.
Continuity mistake: When Kent's father pulls over his car, he holds the steering wheel with his right hand on the top. From the side, barely a frame before, he has his left hand on the side.
Continuity mistake: The first wide shot of Superman arriving at the broken railway shows no logs hanging. The close shots, and then on, do show logs.
Continuity mistake: The way the bags are positioned in the Kent's pickup differs between the side and back angles.
Continuity mistake: When Lois exits the alley where Clark grabs the bullet, her hair is much shorter than when she entered.
Continuity mistake: When Superman arrives at the ditch by the railway, the gap is 3 times his length. In the next shot the gap has shrunk and now measures exactly Superman's height.
Continuity mistake: In Lex Luthor's lair, the bottles to the left of the TV broadcasting the news disappear in the following shot.
Continuity mistake: When Superman is in Luthor's lair and Lex calls Ms. Teschmaker , the lamp on the floor changes from lit to unlit depending on the shot.
Answer: He wanted to see for himself if the stories were true. Some reporters tend to exaggerate the facts and if he had any other weaknesses. He couldn't be sure the kryptonite would work.
While I don't exactly disagree that Lex wanted to see for himself how invincible Superman is, I don't think that's the main reason why he did it. The bullets and fire were simply a charade to make it look like this was his security system. Keep in mind, he wanted Superman to enter his lair because the real trap was the Kryptonite that he had in the lead box.
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