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Also, in comparison to mortal cells, the populations of immortal cells would bloom far more rapidly, causing a resource contention which would swamp mortal cells. This 'fight for food' would negatively affect the immortal cells too, but it wouldn't necessarily cause their extinction as not all cells would starve.
While death is considered a natural part of life, there is no reason why we should die, aside from the fact that at a certain point in one’s life, in the 20s, the regeneration of cells is outpaced by the death of cells, leading us to ageing.