Yes. No question about it.
As an aside, immortality is a supposition that death can occur, but only through such things as injury. Natural lifespan, free of disease and injury, is forever. Invulnerability suggests not even those can kill someone. This idea of living forever, but decaying or injured or diseased and in constant pain, is daft, because it is exactly the things which cause those states that kill people and, consequentially, to live forever those things would have to be prevented or cured.