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 $R$)
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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 $R$.