If you trace Eldarion's descent through his great-grandparents (Earendil, Elwing, Celeborn, Galadriel, Dirhael, Ivorwen, Arador, Unnamed Mortal) he is half-elven.
Through his grandparents he is half-elven because Elrond is an elf.
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"Show it to me"
Thingol to Beren
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