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.