Since skin continuously replaces itself, why do tattoos last a lifetime ?

Since skin continuously replaces itself, why do tattoos last a lifetime ?

The upper layer of the skin, the epidermis, which consists of flattened dried-out dead cells, is replaced continuously throughout life. As cells are worn away from the top layer of skin they are replaced by new cells from below. Tattoos are injected deeper into the skin into a layer called the dermis below where the cells that make the epidermis come from. This means that the dye stays where it is put for life.
Show featuring dermatologist Jane Sterling about the skin and skin diseases