Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 1

Theorem
Let $p$ be a prime number.

Let $b \in \Z_{\ne 0}$:
 * $b, p$ are coprime

Then:
 * $\forall n \in \N: \exists c_n, d_n \in \Z:$
 * $c_n b + d_n p^n = 1$

Proof
Let $n \in \N$.

From Integer Coprime to all Factors is Coprime to Whole:
 * $b, p^n$ are coprime

From Integer Combination of Coprime Integers:
 * $\exists c_n, d_n \in \Z : c_n b + d_n p^n = 1$