For exercise #4 in section 13.3 of the Notes, the definition of a subsequence of a sequence should require that g be strictly increasing on Z+ rather than just nondecreasing. The idea is that the sequence of odd numbers 1, 3, 5, 7, 9,... counts as a subsequence of the sequence of integers 1, 2, 3, 4, 5, 6,..., but the constant sequence 1, 1, 1, 1, 1,... does not.
The problem is easier as written, but not too interesting. The main idea for how to do 13.3.4 in any case is to mimic the subdivision argument we used when proving Heine-Borel. Keep in mind that if you divide infinitely many objects among finitely many boxes, at least one box must contain infinitely many objects.