Real Dictionary

undermine
verb: If you undermine someone or undermine their position or authority, you make their authority or position less secure, often by indirect methods.
verb: If you undermine something such as a feeling or a system, you make it less strong or less secure than it was before, often by a gradual process or by repeated efforts.

undermine in American English
to dig beneath; excavate ground from under, so as to form a tunnel or mine
transitive verb: to injure or destroy by insidious activity or imperceptible stages, sometimes tending toward a sudden dramatic effect

undermine in British English
(of the sea, wind, etc) to wear away the bottom or base of (land, cliffs, etc)