If anyone can help me with this problem, would be greatly appreciated.
Let b be with b > 0 . Show that there exists a Cauchy sequence of Rational numbers { } with such that > 0 for all n of N and that
If is rational then there is nothing to prove - why?
Thus, it is safe to assume is irrational. The for any there exists a rational number so that . Define a sequence . Then it turns out this sequence is Cauchy with .
So what? There are two possibilities. Either it is rational or it is irrational. If rational there is nothing to prove. If irrational then we use the fact that between any two irrational numbers there is a rational number - that is the second case.