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

Use uncountability by interleaving prescribed binary coordinates. For each arbitrary binary sequence $b=(b_k{)}_{k\ge 1}$, define another binary sequence byThere are infinitely many zero coordinates, namely $3k-2$, so $x\in X$. There are also infinitely many coordinates agreeing with the prescribed binary expansion, namely $3k-1$, so $x\in Y$. The coordinates $3k$ recover $b$, making this map an injection of the uncountable set of all binary sequences into $Y$. If $Y$ were a countable set, composing with an injection into $\mathbb{N}$ would make all binary sequences countable, contrary to part (i). Therefore $\boxed{Y\text{ is uncountable}.}$ The construction works for every prescribed binary sequence $(a_n)$; no special distribution property of the digits of $\sqrt{2}$ is needed.