Explain what it means for a set to be countable. Prove that a countable union of countable sets is countable, and that the set of all subsets of is uncountable.
A function is said to be increasing if whenever , and decreasing if whenever . Show that the set of all increasing functions is uncountable, but that the set of decreasing functions is countable.
[Standard results on countability, other than those you are asked to prove, may be assumed.]