Arwen was not born an elf. Tolkien even says she's not an elf. She simply had the aging patterns of one until the time for her choice came. Letter No. 345:
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Arwen was not an elf, but one of the half-elven who abandoned her elvish rights.
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Dior was half-elven:
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their mother being Elwing daughter of Dior, son of Beren and Lúthien: so the problem of the Half-elven becomes united in one line.
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By the marriage of Earendil to Elwing daughter of Dior son of Beren the lines of the Pereldar (Peredil) were united.
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Elwing is half-elven by way of her father who was half-elven as a product of Luthien and Beren.
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Earendil was thus the second of the Pereldar (Half-elven), the elder being Dior, son of Beren and Luthien Tinuviel daughter of King Elu Thingol.
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And most completely, Dior himself says it:
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Dior their son, it is said, spoke both tongues: his father's, and his mother's, the Sindarin of Doriath. For he said: 'I am the first of the Peredil (Half-elven), but I am also the heir of King Elwe, the Eluchil.'
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