I am reading through a textbook on Analysis and have come across a question that I can't seem to make any headway with. A proof is outlined, but I can't make any sense out of it.
The problem is as follows: Let $n$ be a natural in $E^{1}$, and $p,a>0$ be elements of an ordered field $F$. Prove that if $p^{n}>a$, then $(\exists x \in F)|p>x>0$ and $x^{n}>a$.
This is the proof in the book: Let $x=p-d$ with $0
This is where I run into trouble. The Bernoulli inequality states that $\frac{1}{p^n}(1-\frac{d}{p})^{n} \le\frac{1}{p^n}(1-\frac{nd}{p})$. This is fine, but the proof then makes the following step:
$\ldots\implies (1-\frac{d}{p})^{n} \ge (1-\frac{nd}{p})>\frac{a}{p^n}$
Here is my problem - I can't see why the Bernoulli part goes between the other two terms. If anyone could explain it would be much appreciated.