another important comment, immortality isn't related to the land, but to the inner nature.
The elves were inmortal whether they lived in Aman or Middle Earth.
Hobbits or men are mortal and they remain mortal, no matter where they are.
That's clear in the history of Numenor, the kings believe that by arriving to Aman they would become inmortal, but that wasn't true as their nature, as given by Iluvatar was to be mortal.
Frodo and Bilbo's lives were prolongued because they wore the ring, but they couldn't become inmortal, that was something that the Valar couldn't give him, only Iluvatar had the power for that.
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