Before she leaves for America, Eilis looks to her older sister Rose as an example of what a real woman should be. And it's true that Rose is a stone cold fox. But this stone cold fox has been keeping a big secret from the family: she has a terminal heart condition. It's not until this heart condition strikes, ending Rose's life tragically early, that Eilis fully appreciates what her sister sacrificed for her.
Let's start with the happy stuff first. When we meet her, Rose is a woman in full possession of herself, with a steady office job, an ever-expanding list of interested fellas, and a reputation as a killer golf player. In other words, she's everything that Eilis wants to be when she grows up. But there's a darker side to this: Eilis' departure to Brooklyn means that "Rose would not be able to marry" because "it was obvious that their mother could never be left to live alone" (1.222).
This sacrifice makes Rose's death all the more tragic. Rose knows that Eilis won't be able to go to Brooklyn if everyone knows that she's sick: their mom, a widow, would want Eilis to stay and take care of her in her old age if she found out. In fact, Rose probably could have gone to America herself, but chooses instead to give this once-in-a-lifetime opportunity to her little sis. That's a selfless sister, folks. Sadly, Eilis doesn't fully appreciate this sacrifice until Rose's sudden death.
Eilis is devastated upon hearing the news and, after returning to Ireland to help out their mom, gets caught in a swirl of conflicting emotions. These emotions are compounded by their mom, who dresses Eilis in Rose's clothes and makes her feel like "she was Rose's ghost, being fed and spoken to in the same way at the same time by her mother" (4.102). Mrs. Lacey even gets Eilis a temp job at Rose's old office.
Ultimately, however, Eilis realizes that the best tribute to Rose isn't to steal her identity—it's to continue living the way she taught her. After all, Rose sacrificed a lot so that Eilis could give Brooklyn a shot. She kept her illness a secret. She took care of their mom by herself. And she supported Eilis every step of the way. Because of this selflessness, Eilis owes it to her sister's memory to keep boldly going where no small town Irish girl has ever gone before.