Case Closed

When Conan is solving the case he states that the killer smashed the window out. While he was actually running down the hallway when it happened.

To the last comment, I'm pretty sure that Jimmy did hear Rachel say his name, but he probably thought she was dilirious and too drowsy to think straight or remember. When Rachel mentions the incident again at the end of the episode, Jimmy is horrified that she remembered what she had thought and believed he was really there.

Isn't it weird that Jimmy(Conan) doesn't hear Rachel say I missed you Jimmy and I love you until Rachel says that Jimmy saved her from drowning at the end of the second part? I mean, Rachel was almost killed and he asked "Do you know who did it?" and he doesn't hear Jimmy or I love you...He has bad listening skills.