Halcon:
As you explained, thereafter God took away the tree of life...and both Adam and Eve died, just as God told them they would if they ate from the wrong tree.
Maybe you’re reading a different story. In the version I have, God doesn’t tell Eve (who doesn’t get a name until later in the story) anything about either tree, and only tells Adam about the tree of knowledge. The story doesn’t state whether Eve is directly told by God about the tree of knowledge or whether it’s via Adam, but it doesn’t say either are told about the tree of life. God tells Adam he’ll die if he eats from the tree of knowledge, but isn’t told that this was actually a threat about preventing access to the other magical tree rather than a property of the knowledge fruit itself. As presented in the story, the snake didn’t lie when it said that eating the knowledge fruit doesn’t itself cause death, nor does the story say whether the snake knew about the deceptive scheme to prevent access to the other magical tree. Death was caused by the capricious deity getting angry at the naive people for being naive and preventing access to a different magical tree, not at the moment of eating the magical fruit that gave them knowledge.