A look at a few Tripos questions VII

The obligatory question on countability/uncountability.

5C. Define what is meant by the term countable. Show directly from your definition that if is countable, then so is any subset of .

Show that is countable. Hence or otherwise, show that a countable union of countable sets is countable. Show also that for any , is countable.

A function is periodic if there exists a positive integer such that, for every , . Show that the set of periodic functions is countable.