I believe (please correct me if i'm wrong ^_^; ) that Arwen gave up her immortality before or around the time she married Aragorn. Although she was mortal now, she still outlived Aragorn because she was still an Elf, just not an immortal one. Once Aragorn had died she became lonely and eventually wandered off into Lorien to die (I think. Man I'm not good at this kinda stuff)
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