The title of "Peredhel" or "Half-elven" was given only to Elrond and Elros, even though they weren't technically Half-elven. All other children of half-elven unions were mortal. Those two were different because of what Earendil did, and the choice given to him. Elrond chose to become an Elf, and thus, was numbered among the Elves. Elros chose to become a Man, and thus, he and the rest of his line became Mortal. Elrond's children were also given the choice, and we know that Arwen chose to be Mortal. That would mean that all of her children would be mortal as well, especially since she married a Man, Aragorn.
Ergo, Eldarion would be mortal, and not numbered among the Half-elven.
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But Melkor also was there, and he came to the house of Fëanor, and there he slew Finwë King of the Noldor before his doors, and spilled the first blood in the Blessed Realm; for Finwë alone had not fled from the horror of the Dark.
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