Return of the King
"A gift I will give to you. For I am the daughter of Elrond. I shall not go with him now to the Havens; for mine is the choice of Luthien and as she so have I chosen, both the sweet and the bitter. But in my stead you shall go, Ring-bearer, when the time comes and if you then desire it."
So Frodo does take Arwen's place, whether he would otherwise have a right to go or not (which I agree, he does).
Also I think this passage explains Arwen's choice very well, she has chosen "both the sweet and the bitter..." she can not have the best of both worlds.
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"But as for us, we are Hobbits of the Shire, far to the North and West, beyond many rivers."
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