Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 8

Theorem

Let $p$ be a prime number.

Let $a \in \Z$

Then:

$\ds \lim_{n \mathop \to \infty} \dfrac a {p^{n+1}} = 0$

Proof

From Sequence of Powers of Rational Number less than One:

$\ds \lim_{n \mathop \to \infty} \dfrac 1 {p^n} = 0$

From Multiple Rule for Sequences:

$\ds \lim_{n \mathop \to \infty} \dfrac a p \cdot \paren{\dfrac 1 {p^n} } = \dfrac a p \cdot 0 = 0$

The result follows.

$\blacksquare$