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

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

irrational. justification?

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

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

No seems very obvious.

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

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

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

The answer to the limit is five, although I use the fact that you want the limit of the $\displaystyle n$-norm in Euclidean 3-space of the vector $\displaystyle (3,5,2)$ which is the $\displaystyle \infty$-norm of $\displaystyle (3,5,2)$ which is $\displaystyle max \{2,3,5 \}=5$. 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 $\displaystyle \mathbb{R}$ is complete