To be fair God said they would die if they ate the fruit. The fruit did not kill them, in fact the bible even shows that if God had let them continue eating from the tree of life the would have still lived forever. So eating the fruit did not bring sin into their body and their body did not start to die, it was when God would not let them eat from the tree of life.
Genesis 3:22 And the LORD God said, "The man has now become like one of us, knowing good and evil. He must not be allowed to reach out his hand and take also from the tree of life and eat, and live forever