At that point, Frodo hasn't any Admir left, and i think even his Estel is dwindling very fast. He is now pulling hope out of all things wich are near to him, jsut to have a reason not to surrender himself. (thanks to the estel he has left)
even the hope to save gollum after the Ring is destroyed, (and indirectly, the hope he himself can be healed) is a little hope that can be added to his Estel, to be able to hold out a day longer.
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well, that's that then
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