He wasn't able to give them unending life, he only trapped their souls and bound them to the rings. The nine rings were controlled by The One which was part of Sauron, so as long as he was "alive" he held sway over them. But without his will or command they wouldn't have been able to "function" in the world at all. Maybe Sauron could "delay" their aging and deaths by binding their spirits to the wraith world. When Sauron died, their spirits passed away to Mandos.
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Aurë Entuluva!
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