to deal falsely: to lie or deny the truth, put on a false front, or (of a thing) fail to deliver what it promised
This verb draws a line between what is shown and what is real. At its plainest it means to lie or to deny, to say something the speaker knows is untrue, whether to cover a wrong (a neighbor cheated over a deposit) or to disown a fact outright (Sarah's flat denial that she laughed).
From there the word reaches in two directions. Turned toward God, it becomes the language of renouncing him: to deny the LORD, to disown the covenant loyalty owed to him, is to deal falsely at the deepest level. Turned toward a conqueror, it describes beaten enemies who come cringing and swear loyalty they do not mean, a submission that is real on the surface and false underneath.
The same idea stretches to things that let you down. A crop that fails, an olive harvest that comes to nothing, a body wasting away, all break their implied promise and so are said to prove false. Across every use the thread is the gap between appearance and reality, which is why one Hebrew word covers ground that English splits among lie, deny, fail, and cringe.