Davie has to do something about the fact that he's broke and apparently homeless. He has no choice but to go out into the world, so he takes his friend Mr. Campbell's advice and heads to the house of Shaws near Edinburgh to seek help.
Davie arrives at his uncle's house and is disappointed, to say the least, by his unfriendly reception. But there is a silver lining that we think can count as "initial success": after Ebenezer attempts to murder Davie, Davie feels that he has gotten the upper hand. He is sure that Ebenezer is trying to get rid of him because Davie is the true heir to the house of Shaws. At this point, Davie is "all swollen with conceit" (5.4), and in his overconfidence, he really believes that Ebenezer will bring Davie to his lawyer, Mr. Rankeillor, to hand over his estate the next day.
You know that expression, "pride goes before a fall?" Well, all of that conceit leads Davie right into a trap where everything goes wrong. Just when he's at his highest point, sure that he's going to make a ton of money and put everything right, Ebenezer tricks Davie into going on board the Covenant. Davie is kidnapped, threatened with slavery, joins forces with an outlaw, gets shipwrecked, and becomes a murder suspect all due to this one initial bout of overconfidence.
This stage is the moment when the hero is really supposed to come of age, when all his new experiences set him up for the final resolution of his rags-to-riches story. When Davie's on the run with Alan, he learns the traits of self-reliance, loyalty, and friendship. Without this trek across the Highlands, Davie would never have won Alan Breck's friendship and support, which proves instrumental in getting Ebenezer to confess his kidnapping efforts in front of Rankeillor.
We don't know if we'd go so far as to say Kidnapped has "final completion and fulfillment," but Davie does indeed succeed in recovering at least part of his fortune. He's gone from being a penniless orphan to a wealthy laird over the course of around two hundred and fifty pages. That sounds like a rags-to-riches story to us (with some digressions into Scottish history here and there).