well not really.according to the logic behind immortality, your body ages at a very slow rate. Like in trees. Wee human renew our cells every 7 odd years. Trees and longer living beings will have a recycle period with a much longer duration. The timing of aging is controlled in the genes of the species. An immortal being, supposedly, doesn't have a cell breakdown limit, i.e. its cells never lose vitality. Or you can say, regeneration of cells is extremely efficient.

Hence they can persevere.