Prove that the set of all infinite bitstrings is uncountable. Solution We used diagonalization to show that the set of real numbers in the interval [0, 1] is uncountable, i.e. uncountably infinite. The technique was a follows: (1) Assume that a set S can be enumerated. (2) Consider an arbitrary list of all the elements of S. (3) Use the diagonal from the list to construct a new element t. (4) Show that t is in S but is different from all elements in the list and so is not in the list. Contradiction. This shows that the original assumption that S is countable was false, so S is uncountably infinite..