The play opens one year after the suicide of Rosmer's wife, Beata. Rebecca had previously moved into the family home, Rosmersholm, as a friend of Beata, and she lives there still. It becomes plain that she and Rosmer are in love, but he insists throughout the play that their relationship is completely platonic.
A highly respected member of his community, Rosmer intends to support the newly elected government and its reformist, if not revolutionary, agenda. However, when he announces this to his friend and brother-in-law Kroll, the local schoolmaster, the latter becomes enraged at what he sees as his friend's betrayal of his ruling-class roots. Kroll begins to sabotage Rosmer's plans, confronting him about his relationship with Rebecca and denouncing the pair, initially in guarded terms, in the local newspaper. Rosmer becomes consumed by his guilt, now believing he, rather than mental illness, caused his wife's suicide. He attempts to escape the guilt by erasing the memory of his wife and proposing marriage to Rebecca. But she rejects him outright. Kroll accuses her of using Rosmer as a tool to work her own political agenda. She admits that it was she who drove Mrs. Rosmer to deeper depths of despair and in a way even encouraged her suicideâinitially to increase her power over Rosmer, but later because she actually fell in love with him. Because of her guilty past she cannot accept Rosmer's marriage proposal.
This leads to the ultimate breakdown in the play where neither Rosmer nor Rebecca can cast off moral guilt: she has acknowledged her part in the destruction of Beata but she has also committed incest with her supposedly adoptive father while suspecting that he was in truth her natural parent. Her suspicion is harshly confirmed by Kroll when he attempts to come between her and Rosmer; they can now no longer trust each other, or even themselves. Rosmer then asks Rebecca to prove her devotion to him by committing suicide the same way his former wife didâby jumping into the mill-race. As Rebecca calmly seems to agree, issuing instructions about the recovery of her body from the water, Rosmer says he will join her. He is still in love with her and, since he cannot conceive of a way in which they can live together, they will die together. The play concludes with both characters jumping into the mill-race and the housekeeper, Mrs. Helseth, screaming in terror: "The dead woman has taken them".

Question: How does Rosmer want Rebecca to prove her love?
Answer: Rosmer wants her to commit suicide like Ms. Rosmer did. 
Alternative answer: by committing suicide