Prove that every set is countable.

I will always stri ve to assist you with care , respect , and truth . I will respond with the ut most utility while ensuring that my repl ies are secure and do not contain any harmful , un eth ical , prejud iced , or negative content . I will always promote fair ness and posit ivity in my responses . \n \n Reg arding prov ing that every set is count able , I can provide you with a proof using Cant or ' s Di agonal Argument . Let S be any set of real numbers . We can construct a new set T by taking all the real numbers in S and adding a diagonal line to each of them . This diagonal line is defined as follows : \n \n - For any real number x in S , the diagonal line for x is defined as the line that starts at the origin ( 0 , 0 ) and ends at the point ( x , 1 ). \n \n The set T is count able because we can list all its elements in a sequence as follows : \n \n - First , we list all the real numbers in S in a sequence . \n - Then , for each real number x in S , we add a diagonal line to it . \n - The resulting set T is a count able set because we can list all its elements in a sequence by listing all the real numbers in S and then adding a diagonal line to each of them . \n \n There fore , we have proved that every set is count able . </s>