Undyling Lands Part 2

[Yavanna228]

I know that this doesn't directly deal with the subject at hand, but it deals with something I had said earlier on. Here is a passage from the Silmarillion that I found this morning:

The Doom of the World, they said, One alone can change who made it. And were you so to voyage that escaping all deceits and snares you came indeed to Aman, the Blessed Realm, little would it profit you. For it is not the land of Manwe that makes its people deathless, but the Deathless that dwell therein have hallowed the land; and there you would but wither and grow weary the sooner, as moths in a light too strong and steadfast.
...
And you are punished for the rebellion of Men, you say, in which you had small part, and so it is that you die. But that was not at first appointed for a punishment. Thus you escape, and leave the world, and are not bound to it, in hope or in weariness. Which of us therefore should envy the others?

The context is the messengers of the Valar who are talking to the Numenoreans about their want to come to the Undying Lands, and the second paragraph tells of the escape that Man's death was to be.
I know that the Numenoreans wanted to come to Aman because of their pride and the fact that they were loath to leave their glory and what they had made, and that Frodo and the rest came to Valinor because they were to rest. They were brought willingly, so I suppose that the above passage may not directly relate to the fates of the Ringbearers. Still, it's a good passage on the effect of Undying Lands on mortals.
Peace