I would like some help with this problem. I've been given that we have a $k^p$ s.t. $p$ is irrational and $k$ is a positive integer satisfies the following:
- $k^r\lt k^p$ if $r$ is positive and rational while $p$ is positive and irrational and $r\lt p$. and
- $k^r \gt k^p$ if $p\lt r$ for positive $p$ and $r$.
I need to derive the $p$ test for $p$ being irrational by using the following:
- validity of the $p$ test when $p$ is rational
- Comparison test for series
- Denseness of rationals.
How would I go about doing this? What I have done is that I have taken $k^p$ and $k^r$ both their inverses because this is the only way we could use the comparison test. But beyond this I'm not sure where any of it leads to.
