The man will realize the wisdom of the old-timer’s warning.
Explanation:
"To build a fire" is a short story by Jack London, this story is about a man who travels to the Yukon ignoring the warnings from an old-timer about the deathly Yukon winters (where temperatures can reach 40 below)
In this excerpt we can see the man saying he was safe, remembering the old-timer warning and thinking that the old-timers were "rather womanish". We can see that he thinks that the warning was not real and that he was safe and will be able to survive and deal with the subzero temperatures of the Yukon, thus, the best prediction based on this excerpt is that he will likely won't be able to continue with this streak of "safeness" and he will realize the wisdom of the old-timer's warning.
