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

Theorem
Let $p$ be a prime number.

Then:
 * $\exists x \in \Z_{\gt 0}: p \nmid x, x \gt \dfrac {p+1} 2$

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

Then:

and

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