Only one thing to add. Galadriel did not give the Elf-stone to Aragorn when he became king of the united kingdoms, she Gave it to him when they last met at Lothlorien. She does say that she had passed it down to her daughter, and she to hers and had been left in Lothlorien for Aragorn should he pass that way. However, Not only was he foretold to be called Elessar, but in this way, it also served as the gift of the bride-to-be's mother to the groom. Elf customs had the mother of the bride giving the groom a gift, and since Celebrian was no longer in Middle-earth, it became Galadriel's duty as Grandmother of Arwen.