This is Theorem 7.5 in Foundations of Mathematical Analysis by Johnsonbaugh and Pfaffenberger.
At the end of the proof in the book, we want to show by contradiction that $b^n < a$ and $b^n > a$ are not true. The proof in the book excludes the part for $b^n > a$. The text suggest it's to be proved in a similar manner to $b^n<a$, but I can't seem to figure it out how to set up the variables to prove $b^n > a$ is not true.
Here is the proof in the book:
My problem lies in "Similarly, one shows that $a < b^n$ is false...". How is this done?