That was the idea. Both proofs are easy enough such that they can be figured out, and the statements are important enough to be learnt.There's really nothing to prove other than the well known theorems
By Modulus maximum principle [itex]f[/itex] must be constant on [itex]U_0\subseteq U[/itex].
By the Identity theorem, since [itex]U[/itex] is open and connected, [itex]f[/itex] must be constant on [itex]U[/itex].
Guess you all recognized that I begin to run out of good ideas. Algebra is usually not touched even if the problems are really easy. Geometry is a horror to check or correct without a common picture (plus that I do not know the correct English names of the objects, e.g. angle bisector). Functional analysis is either well-known, used, or a nightmare about the many categories (e.g. different convergences, Sobolev). Most of the interesting problems require an even more advanced framework than algebra does. Calculations like integrals, ODE, or inequalities are either used up or simply boring. And now that you just quote the theorems or corollaries I want to be proven, this source is exhausted, too. Any ideas?