To use diagonalization to prove that a set is uncountable, you typically do a proof by contradiction: assume that 'is' countable, so that there is a surjection , and then find a contradiction by constructing a diabolical object that is not in the image of . This contradicts the surjectivity of , completing the proof.
To construct infinite) table describing : the row describes . In the table, you represent each element of as an infinite sequence of values; so that the table has a column for each natural number. The exact structure of the table depends on the set ., you typically imagine an (