1. ## Sequences, series, completeness

Some questions that I can't quite get my head around:

1) Show that every sequence which is not bounded above has an increasing subsequence.

2) Show that if $(an) \rightarrow 0$ then there is a subsequence of $(\frac {1} {an})$ which tends to $+ \infty$, or there is a subsequence of $(\frac {1} {an})$ which tends to $- \infty$.

3) If a is irrational and b is not equal to 0 and b is an integer, is $\frac {a} {b}$ rational or irrational? Justify your answer.

irrational. justification?

4) What is the value of $\lim_{n \to \infty} \ (3^n + 5^n+ 2^n)^{\frac {1} {n}}$ ?

5) State, without proof, whether or not there can exist a Cauchy sequences $(a_n)$ in R (real numbers), which does not converge in R (real numbers).

No seems very obvious.

2. If $\{ a_n \}$ is unbounded above, take $b_1= a_1$ and find $b_n$ recursively: Given $\{ b_n \}_{n \leq m}$ there exists $b_m$ such that $b_m \geq b_n$ for all $n \leq m$, because otherwise taking $M= max_{n \leq m}{\vert b_n \vert }$ would be a bound of $\{a_n \}$.

If $\{a_n \} \longrightarrow 0$ then $\{ \frac{1}{a_n} \}$ is unbounded, and by the previous exercise has an increasing or decrasing subsequence.

If $a$ is irrational, $b$ an integer and $q= \frac{a}{b}$ were a rational number, since $\mathbb{Q}$ is a field then $qb$ would be a rational number but $qb=a$. therefore $q$ is irrational.

The answer to the limit is five, although I use the fact that you want the limit of the $n$-norm in Euclidean 3-space of the vector $(3,5,2)$ which is the $\infty$-norm of $(3,5,2)$ which is $max \{2,3,5 \}=5$. If you've seen this I'll elaborate, if not then use this only as a guide to the answer.

And finally, No, there cannot exist Cauchy sequences which do not converge because $\mathbb{R}$ is complete