Suppose is denumerable. Prove that there is a partiion of such that is denumerable and for every is denumerable.

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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 .