Homework Solutions 1.

1.

For any $n\geq 1$ , let $q_{2n-1}=n$ and $q_{2n}=n-\frac{1}{n}$ .
2.

For any $n\geq 1$ , let $q_{n}=n^{2}$ ,
3.

For any $n\geq 1$ , let $q_{2n-1}=n$ , $q_{2n}=\frac{1}{n}$ .
4.

Fix $\varepsilon>0$ . Then we have $N\geq 1$ so that if $n,m>N$ , then $|q_{n}-q_{m}|<\varepsilon/2$ and $|r_{n}-r_{m}|<\varepsilon/2$ . Then $|(q_{n}+r_{n})-(q_{m}+r_{m})|<\varepsilon$ showing that $\{q_{n}+r_{n}\}^{\infty}_{n=1}$ is a Cauchy-sequence.