P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 2

Theorem

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $\Z_p$ be the $p$-adic integers.

Let $x \in \Z_p$.

Let $\sequence {\alpha_n}$ be a sequence of integers:

$\norm {x -\alpha_n}_p \le p^{-\paren {n + 1} }$

Then:

$\ds \lim_{n \mathop \to \infty} \alpha_n = x$

Proof

From Sequence of Powers of Number less than One:

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

From Multiple Rule for Real Sequences:

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

By the Squeeze Theorem for Real Sequences :

$\ds \lim_{n \mathop \to \infty} \norm {x - \alpha_n}_p = 0$.

Hence the limit of $\sequence {\alpha_n}$ is by definition:

$\ds \lim_{n \mathop \to \infty} \alpha_n = x$

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.4 \ \text {(ii)}$