In Lord of the Rings: The Fellowship of the Ring, Aragorn, as Strider, tells the four Hobbits about the Nazgûl, and describes them as "neither living nor dead." Gollum later tells Sam and Frodo while they're trekking through the Dead Marshes that the Nazgûl cannot be killed. Yet, during the battle at Minas Tirith, Éowyn simply stabs the Witch King of Angmar in the face with her sword and he ceases to be.
How could Éowyn have slain the Witch King of Angmar if he wasn't alive to begin with, and was supposedly unable to be killed, as he was caught between life and death in some kind of Nazgûl-ish limbo? How did she destroy him?