Paper 4, Section II, E

Numbers and Sets | Part IA, 2009

Prove that the set of all infinite sequences (ϵ1,ϵ2,)\left(\epsilon_{1}, \epsilon_{2}, \ldots\right) with every ϵi\epsilon_{i} equal to 0 or 1 is uncountable. Deduce that the closed interval [0,1][0,1] is uncountable.

For an ordered set XX let Σ(X)\Sigma(X) denote the set of increasing (but not necessarily strictly increasing) sequences in XX that are bounded above. For each of Σ(Z),Σ(Q)\Sigma(\mathbb{Z}), \Sigma(\mathbb{Q}) and Σ(R)\Sigma(\mathbb{R}), determine (with proof) whether it is uncountable.

Typos? Please submit corrections to this page on GitHub.