Why do you think the serpent didn’t suggest Eve to also take fruit from the tree of life?

The serpent was the enemy of God.  He came to deceive Adam and Eve and ultimately to destroy what God cherished so much.  He lied to Eve so that they would disobey God, and that was the one thing that God told them not to do.

The tree of life is inconsequential to their wills and obedience, and that was what was at stake.