Homework 1.

1. Give an example of a sequence of positive rational numbers ${\{q_n\}}_{n=1}^{\mathrm{\infty }}$ such that the following statement holds: $\forall \epsilon >0$, $\exists n>1$ such that $q_n>\frac{1}{\epsilon }$ and .

   (2 points)

2. Give an example of a sequence of positive rational numbers ${\{q_n\}}_{n=1}^{\mathrm{\infty }}$ such that the following statement holds: $\forall K>0$, $\exists n>1$ such that $|q_n-q_{n+1}|>K$.

   (2 points)

3. Give an example of a sequence of positive rational numbers ${\{q_n\}}_{n=1}^{\mathrm{\infty }}$ such that the following statement holds: $\forall$ positive rational number $a$ there exists $n\ge 1$ such that $q_n>a$ and .

   (2 points)

4. Show that if ${\{q_n\}}_{n=1}^{\mathrm{\infty }}$ and ${\{r_n\}}_{n=1}^{\mathrm{\infty }}$ are Cauchy-sequences of rational numbers, then ${\{q_n+r_n\}}_{n=1}^{\mathrm{\infty }}$ is a Cauchy-sequence as well.

   (4 points)

In the first three problems define the examples in a clear fashion. (you do not necessarily need to give an exact formula for the sequences)