non constant harmonic function

Question

If $u$ is harmonic function on disk with radius $R$ around the origion, and non constant in it.

why is it true that $u$ cannot be constant in any sub-Disk (i.e disk with radius less than $R$)

thanks

Answer 1

Because than $u$ is equal to a constant function on an open set (the sub disk) which implies by the identity theorem for harmonic function that $u$ is constant on the disk with radius $R$.