(c) Let $A_n$ be an increasing sequence of sets, and $B_n$ a decreasing sequence of sets. Set $A = \bigcap_{n=1}^{10} A_n$ and $B = \bigcap_{n=1}^{10} B_n$. Give a Simple expression for $T = A \cup B$.
(d) Let $a, b, c$ and $\alpha, \beta$ be positive numbers such that $a > b > c$ and $\alpha > \beta$. Give a simplified expression for the limit $L$ of $\frac{x^\alpha - x^{\beta/3}}{x^b - x^\alpha + x^{c/2} - \alpha}$ as $x$ tends to $\infty$.
(e) Let $a_n = (-1)^{n+2}, b_n = (-1)^{n+3}, c_n = 1/(-1)^{n+1}, S_n = (a_n+b_n), P_n = (a_n-b_n)/c_n^2$. Give a simplified expression for $S_n$ and for $P_n$, where $n \in \mathbb{N}$.
(f) Let $H = (A \setminus B) \cup C$, where $A = (0,7), B = (6,7] \cup \{3,4\}$, and $C = \{2,4,8,9\}$. Let $W = \{x \in \mathbb{R}: H \text{ is a neighborhood of } x\}$. Give $W$ explicitly.
(g) Let $S$ be a bounded subset of $\mathbb{R}$. Let $m = \inf S, M = \sup S$, an let $\varepsilon > 0$. Give a number $P$ depending on $m$ and $\varepsilon$ and a number $Q$ depending on $M$ and $\varepsilon$ such that $P$ is not a lower bound of $S$ an $Q$ is not an upper bound of $S$.
(h) Let $C = \{8 + \frac{3(-1)^{n+5}}{n} : n \in \mathbb{N} \text{ and } n \text{ odd} \}$. Give $\inf(C)$ and $\min(-C)$, where $-C = \{-x: x \in C\}$.
