Consider how the diagonal argument constructs a new element that differs from every element in a given infinite list, and why this implies the list cannot be exhaustive.

Explore the implications of the diagonal construction when applied to the set of all infinite binary sequences, and how this leads to a contradiction if we assume such a set is countable.

Formalize the diagonal argument by defining a function that maps natural numbers to sequences and then construct a sequence that cannot be in the range of this function, thereby proving the uncountability of the set of infinite binary sequences.