P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 1

Theorem
Let $p \ge 3$ be a prime number.

Let $b = \dfrac {p + 1} 2$ Then:
 * $b \in \Z: 0 \lt b \lt p$

Proof
Since $p$ is odd then $p + 1$ is even and $b \in \Z$

Then:

and

Hence:
 * $0 \lt b \lt p$