91Ó°ÊÓ

For each sets below determine if it is bounded above, bounded below, or both. If it is bounded above (below) find the supremum (infimum). Justify all your conclusions. (a) \(\left\\{\frac{3 n}{n+4}: n \in \mathbb{N}\right\\}\) (b) \(\left\\{(-1)^{n}+\frac{1}{n}: n \in \mathbb{N}\right\\}\) (c) \(\left\\{(-1)^{n}-\frac{(-1)^{n}}{n}: n \in \mathbb{N}\right\\}\)

Let \(A\) be a nonempty subset of \(\mathbb{R}\) and \(\alpha \in \mathbb{R}\). Define \(\alpha A=\\{\alpha a: a \in A\\}\). Prove the following statements: (a) If \(\alpha>0\) and \(A\) is bounded above, then \(\alpha A\) is bounded above and \(\sup \alpha A=\alpha \sup A\). (b) If \(\alpha<0\) and \(A\) is bounded above, then \(\alpha A\) is bounded below and inf \(\alpha A=\alpha \sup A\).

Prove that in between two real numbers \(a\) and \(b\) with \(a