$1.$We define a sequence of rational number {$a_n$} by putting $a_1 =3,\;\text{ and}\;\; a_{n+1} = 4 - \frac{2}{a_n} \text{ for all}\; n \in \mathbb{R}.\;\text{ Put}\;\; \alpha = 2 + \sqrt{2}.$
$(a)$ Calculate $a_1,\ a_2,\ a_3,\ a_4,\ a_5$, and $a_6.$ Determine the decimal expansion of $a_6$ and $\alpha$ on your calculator.
$(b)$ Prove, by induction on $n$ that $3 \leq a_n \leq 4$ for all $n \in \mathbb{N}.$
$(c)$ Show that $\displaystyle 3 \leq \alpha \leq 4$ and $\alpha = 4 - \frac{2}{\alpha}$.
$(d)$ Show that $\displaystyle a_{n+1} - \alpha = \frac{2(a_n-\alpha)}{\alpha a_n}$ for all $n \in \mathbb{N}.$
$(e)$ Prove, by induction on $n$, that $\displaystyle |a_n - \alpha| \leq \frac{|a_1 - \alpha|}{4^{n-1}}$ for all $n \in \mathbb{N}$.
$(f)$ Deduce that $a_n \rightarrow \alpha$ as $n \rightarrow \infty$.
$2.$Find a $rational$ function $f: \mathbb{R} \longrightarrow \mathbb{R}$ with range $f(\mathbb{R}) = [-1,\ 1].$ (Thus $\displaystyle f(x) = \frac{P(x)}{Q(x)}$ for all $x \in \mathbb{R}$ for suitable polynomials $P$ and $Q$ where $Q$ has no real root).
$1(a-d)$ are completed but $1(e)$, $1(f)$ and $2$ are still confusing me.