Forever, Damon Knight, 1981

In 1887, the secret of immortality is discovered and administered to the Gerd Essenwien, the 12-year-old son of the discoverer. It becomes apparent that it also stops aging as well, trapping him in a boy’s body. Shortly after this event, a serum is discovered to stop all bacteriological disease, a universal vaccine. Both are adopted widely and having children falls out of fashion. People still die of accidents and heart disease and cancer, so the population of the Earth begins to dwindle. When humanity panics over the looming end of the human race, they find that the youngest women are still over two hundred years old and infertile.

Gerd Essenwein is eventually the last man on Earth, keeping in touch via shortwave with the last woman on Earth, a Japanese woman with the apparent age of 16. She travels by bicycle to Europe so they can meet (it takes her eleven years.) When he shows her that he cannot perform sexually, she returns to Japan and disappears from shortwave contact a few years later. Aliens arrive and ask Gerd to come with them when they find he is the last man on Earth. Gerd refuses and they leave.

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