P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 5

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Theorem

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$.

Let $k$ be any positive integer.

Let $\map f X \in \Z \sqbrk X$ be the polynomial:

$X^k - a$

for some $a \in \Z$.

Let $\sequence {x_n}$ be a sequence of integers such that:

$\forall n: \map f {x_n} \equiv 0 \pmod {p^n}$


Then:

$\displaystyle \lim_{n \mathop \to \infty} x_n^k = a$ in $\struct {\Q, \norm{\,\cdot\,}_p}$


Proof

By assumption:

$\forall n \in \N: p^n \divides \paren {x_n^k - a}$

By the definition of the $p$-adic norm:

$\forall n \in \N: \norm {x_n^k - a}_p \le \dfrac 1 {p^n}$

By Sequence of Powers of Number less than One:

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

By Squeeze Theorem for Real Sequences:

$\displaystyle \lim_{n \mathop \to \infty} \norm{x_n^k - a}_p = 0$.

By the definition of convergence in $\struct {\Q, \norm{\,\cdot\,}_p}$ then:

$\displaystyle \lim_{n \mathop \to \infty} x_n^k = a$

$\blacksquare$