Past exam of the mathematics course of the University of Cambridge / 2014 / ia / Paper 4 / 7E / i / Solution

A set is a countable set if it admits an injective function into the natural numbers; this includes finite sets and countably infinite sets. A countably infinite set can be listed by a bijection from $\mathbb{N}$.

Let $Z$ be the set of infinite binary sequences. Suppose it were countable and list all its members as $s^{(1)},s^{(2)},\dots$, writing $s_n^{(j)}$ for the $n$th entry of row $j$. A finite listing could be repeated to make such a list as well. Define the binary sequence $y$ by $y_n=1-s_n^{(n)}$. It differs from the $n$th row in coordinate $n$, for every $n$. Thus it belongs to $Z$ but is missing from the list, a contradiction. The set $Z$ is uncountable, by Cantor's diagonal argument.