I tried to prove the above statement showing that the boundary of each proper set is non-empty. But i could not did it then i tried to construct a line joining each two points in the set Banach space, and show that each such line is connected, to show that each line is connected i tried to prove that each proper set is non-empty using supremum and infimum, i could do this, but whenever the supremum and the infimum of the proper set is one of the points i could not show that the boundary of the proper set is not empty.
How it may be proven?
ps: The space does not need to be complete, furthermore we may assume the scalar field to be either the real or complex numbers.