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 then there is a subsequence of which tends to , or there is a subsequence of which tends to .

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

irrational. justification?

4) What is the value of ?

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

No seems very obvious.