This is a concise question, so the title pretty much says it all. Also, this is not a HW question, but the idea has subtly popped up in two homework problems that I have done in the past. I cannot justify why the entire set would be bounded, because we know nothing of the nature of the boundedness of each element; i.e. the bounds themselves could grow without bound, e.g. if each element a_n is bounded by n, then surely the set is not bounded.