Immortality is a death sentence; the cure is death. The telomeres in our cells are effectively a counter. Each split decrements the counter, preventing the unlimited splitting of cells. Such unrestricted division might occur if the cell mutates to ignore the neighbouring signals that they should stop splitting.

A cell going rogue, dividing indefinitely, is what we call cancer. If not for telomeres it would be a death suffered far more frequently. However, this limit on the number of times a cell can divide is what results in ageing, and eventually death. Telomeres solve the problem of immortal, fatal cells by killing them and subsequently us after some time.