How does chapter 44 offer hope for a happy ending in Pride and Prejudice by Jane Austen?

Elizabeth is on holiday with her aunt and uncle Gardiner in Derbyshire, the county where Mr. Darcy and his sister reside.  Mr. Darcy has asked Elizabeth if he could introduce her to his sister, Georgiana, and it is in this chapter that the two meet for the first time.  The Gardiners watch all of the goings-on between Darcy and Elizabeth with much interest, and "of the lady's sensations they remained a little in doubt; but...

Elizabeth is on holiday with her aunt and uncle Gardiner in Derbyshire, the county where Mr. Darcy and his sister reside.  Mr. Darcy has asked Elizabeth if he could introduce her to his sister, Georgiana, and it is in this chapter that the two meet for the first time.  The Gardiners watch all of the goings-on between Darcy and Elizabeth with much interest, and "of the lady's sensations they remained a little in doubt; but that the gentleman was overflowing with admiration was evident enough."  It is clear to them that Mr. Darcy loves Elizabeth; they only question whether she definitively returns his feelings.  

Further, Mr. Bingley and Elizabeth meet again in this chapter, since Bingley is with Darcy's party at Pemberley, Darcy's estate.  Elizabeth cannot see any signs of romantic interest between Bingley and Miss Darcy, as Miss Caroline Bingley had tried to persuade Jane Bennet to believe.  Mr. Bingley not-so-subtly inquires about Elizabeth's sisters, and she assures him that they are all at home (except Lydia), and it is pretty obvious to her that he still cares for Jane.

Elizabeth, too, is struck by the changes in Mr. Darcy.  Where she was once so set against him, even rejecting his proposal and insulting him, she now feels that "such a change in a man of so much pride, excited not only astonishment but gratitude — for to love, ardent love, it must be attributed."  She begins to take a real interest in his person, his welfare, and she wonders if it is possible that he might renew his proposals to her.  

Thus, the probability of happy prospects for both Jane and Elizabeth give readers reason to hope for — and even expect — a happy ending for the two young women.