How did Murtagh kill Eragon in The Inheritance Cycle?

Well, honey, Murtagh didn't actually kill Eragon in The Inheritance Cycle. In the third book, Brisingr, Murtagh and Eragon have a big ol' fight, but it's their dragons, Thorn and Saphira, who end up getting all hot and bothered and causing chaos. So, no need to go spreading rumors about poor Murtagh being a dragon-slayer.