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 an integer sequence such that:
- $\forall n: \map f {x_n} \equiv 0 \pmod {p^n}$
Then:
- $\ds \lim_{n \mathop \to \infty} {x_n}^k = a$ 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:
- $\ds \lim_{n \mathop \to \infty} \dfrac 1 {p^n} = 0$
By Squeeze Theorem for Real Sequences:
- $\ds \lim_{n \mathop \to \infty} \norm { {x_n}^k - a}_p = 0$
By the definition of convergence in $\struct {\Q, \norm {\,\cdot\,}_p}$:
- $\ds \lim_{n \mathop \to \infty} {x_n}^k = a$
$\blacksquare$