Mortal flaw in narrative -- BIG SPOILER

BIG SPOILER -- DO NOT READ THIS UNLESS YOU HAVE SEEN THE MOVIE

In the narrations that overlay the action the two main characters tell us this story as if it they are reflecting on it, as if it happened to them in the past. But the professor dies at the end. How can he tell us a story that happened to him if he died before that story ended? I was not sure he was dead until the credits started rolling. Actually, I guess, I am still not sure he died.