Workshop Exercises 1.

1.
First try to remember the definitions. Then check your answer using the notes.
- –
  
  State the definition of a convergent sequence in $\mathbb{Q}$ .
- –
  
  State the definition of a Cauchy-sequence in $\mathbb{Q}$ .
- –
  
  What does it mean that a sequence of rational numbers $\{a_{n}\}^{\infty}_{n=1}$ is bounded?
2.

Use your calculator. What is $((\pi)_{6})^{2}$ ? What is $(\pi^{2})_{6}$ ?
3.

Give an example of a sequence of positive rationals $\{a_{n}\}^{\infty}_{n=1}$ such that the following statement holds: $\forall\epsilon>0$ , $\exists$ $n\geq 1$ such that $a_{n}<\epsilon$ and $a_{n+1}>\frac{1}{\epsilon}$ .
4.

Let $\{a_{n}\}$ , $\{b_{n}\}$ be sequences of rational numbers such that $a_{n}\to a\in\mathbb{Q}$ and $b_{n}\to b\in\mathbb{Q}$ . Show that $(a_{n}+b_{n})\to a+b$
5.

Let $a=\{a_{n}\}^{\infty}_{n=1}$ , $b=\{b_{n}\}^{\infty}_{n=1}$ , $c=\{c_{n}\}^{\infty}_{n=1}$ , $d=\{d_{n}\}^{\infty}_{n=1}$ be Cauchy-sequences of rational numbers. Suppose that $a\equiv b$ and $c\equiv d$ . Show that $a+b\equiv c+d$ .
6.

Show that if $x<y$ are rational numbers, then there exist infinitely many rational numbers $\{a_{n}\}^{\infty}_{n=1}$ so that for any $n\geq 1$ , $x<a_{n}<y$ .
7.

Show that for any $\varepsilon>0$ , there exists a number $n\geq 1$ and a finite set $a_{1},a_{2},\dots a_{n}$ such that the following statement holds:

$\forall 0\leq x\leq 1,\exists\,1\leq k\leq n$ such that $|x-a_{k}|<\varepsilon$ .
8.

Give an example of a sequence of positive rationals $\{a_{n}\}^{\infty}_{n=1}$ such that the following statement holds: $\forall x,y$ so that $0<x<y$ , $\exists n\geq 1$ such that $x<a_{n}<y$ .
9.

Give an example of a sequence of positive rational numbers such that for $\forall 0<x<y$ $\forall\varepsilon>0$ , $\exists n\geq 1$ such that $|a_{n}-x|<\varepsilon,|a_{n+1}-y|<\varepsilon$ .