ELI5 HIV/AIDS - undetectable but not cured

ART is able to prevent the virus from replicating making the patient undetectable and in theory unable to transmit the virus. Cells have a life span and would be replaced by uninfected cells, no? How is the virus not eliminated over time?

Edit: thanks for all the responses. I keep Seeing the drug ads on tv and this had me perplexed.