Exercise.

(a) Find the set of all $x\in \mathbb{R}\setminus \left\{-3,-5\right\}$ such that $\frac{x+1}{x+3}<\frac{x+3}{x+5}$.

(b) Find the set of all $x\in \mathbb{R}\setminus \left\{-3,-4\right\}$ such that $\frac{x+1}{x+3}<\frac{x+3}{x+4}$.

(c) Find the set of all $x\in \mathbb{R}\setminus \left\{-3,-2\right\}$ such that $\frac{x+1}{x+3}<\frac{x+3}{x+2}$.

Give your answers as an interval or a union of disjoint intervals.

Exercise.

(a) Find the set of all $x\in \mathbb{R}$ such that $\left|x\right|>{x}^{2}-1$.

(b) Find the set of all $x\in \mathbb{R}$ such that $3\left|x\right|>{x}^{2}+1$.

Give your answers as an interval or a union of disjoint intervals. Leave any numbers in your answers in surd form.

Exercise.

Consider the sequence defined by ${a}_{n}=\frac{\sqrt{n}}{\sqrt{n+1}}$. Prove that ${a}_{n}\to 1$ as follows.

(a) Find as simple an expression as you can for $\left|{a}_{n}-1\right|$. Make sure your expression does not involve absolute value signs.

(b) Supposing $\epsilon >0$ is given, use your answer to (a) to find the set of $n\in \mathbb{N}$ such that $0\u2a7d\left|{a}_{n}-1\right|<\epsilon $ writing the answer as $A\cup \{n\in \mathbb{N}:n\u2a7eF\left(\epsilon \right)\}$, where $A$ is a finite set of naturals numbers and $F\left(\epsilon \right)$ is an expression involving $\epsilon $ only. (Hint: By conjugating surds, show that $\left|{a}_{n}-1\right|<\frac{1}{2n}$ for all $n$. You don't have to say what your set $A$ is. Why is it finite? Be sure to use $\u2a7e$ not $>$. If you have $\{n\in \mathbb{N}:n>F\left(\epsilon \right)\}$ you can replace it with $\{n\in \mathbb{N}:n\u2a7eF\left(\epsilon \right)+1\}$ if you change the set $A$.)

(c) Write down a proof that ${a}_{n}\to 1$ starting with
Let $\epsilon >0$ be arbitrary

and
Let $N=\lceil F\left(\epsilon \right)\rceil $

.

Exercise.

Consider the statement **X**, which is

and the sequence defined by $\left({a}_{n}\right)={-1}^{n}+\frac{1}{n}$.
Prove that **X** holds for this sequence for both $l=1$ and
$l=-1$.

**All assertions you make in your answers MUST be supported by proofs.**