Paper 4, Section II, E

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

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

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