Suppose is denumerable. Prove that there is a partiion of such that is denumerable and for every is denumerable.
Note that for any partition , it is not necessarily denumerable, and it is not necessarily that must be denumerable for all . For example, and
I think I must construct some Note that and must be written as
(A can be written like this since A is denumerable.)
I cannot find any way to construct such , other than what they should be by the characterization above.
Must I resort to proof by contradiction?
Thank you very much.
PS. Definition of partition, not equal .