Problem 11
Prove that if \(x_{n} \leq y_{n}\) for all \(n\), then \(\underline{\lim } x_{n} \leq \underline{\lim } y_{n}\) and \(\lim x_{n} \leq \overline{\lim } y_{n}\).
Problem 13
Prove that if \(A\) consists of positive elements only, then \(q=\sup A\) iff
(i) \((\forall x \in A) x \leq q\) and
(ii) \((\forall d>1)(\exists x \in A) q / d
