Hensel's Lemma/P-adic Integers/Lemma 2
Theorem
Let $\Z_p$ be the $p$-adic integers for some prime $p$.
Let $\alpha \in \Z_p$ be a $p$-adic number with $p$-adic expansion:
- $\alpha = \ds \sum_{n \mathop = 0}^\infty d_n p^n$
Let $\alpha_0 \in \Z_p$ be a $p$-adic integer.
Let $p \Z_p$ denote the principal ideal of $\Z_p$ generated by $p$.
For all $x, y \in \Z_p$, let:
- $x \equiv y \pmod {p \Z_p}$
denote congruence modulo the ideal $p \Z_p$.
For all $k \in \N$, let the partial sum $a_k = \ds \sum_{n \mathop = 0}^k d_n p_n$ satisfy:
- $a_k \equiv \alpha_0 \pmod {p \Z_p}$
Then:
- $\alpha \equiv \alpha_0 \pmod {p\Z_p}$
Proof
From Ideals of P-adic Integers:
By Definition of Congruence Modulo an Ideal:
- $\forall k \in \N: a_k - \alpha_0 \in p \Z_p$
By Definition of $p$-adic Expansion:
- $\alpha = \ds \lim_{k \mathop \to \infty} a_k$
From Sum Rule for Sequences in Normed Division Ring:
- $\alpha - \alpha_0 = \ds \lim_{k \mathop \to \infty} a_k - \alpha_0$
From Closed Subgroups of P-adic Integers:
- $p \Z_p$ is a closed set in the $p$-adic metric
From Subset of Metric Space contains Limits of Sequences iff Closed:
- $\alpha - \alpha_0 \in p \Z_p$
By Definition of Congruence Modulo an Ideal:
- $\alpha \equiv \alpha_0 \pmod {p \Z_p}$
$\blacksquare$