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**Question**

**Consider the sequence [itex]\{p^n\}_{n\in\mathbb{N}}[/itex]. Prove that this sequence is Cauchy with respect to the p-adic metric on [itex]\mathbb{Q}[/itex]. What is the limit of the sequence?**

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- Thread starter Oxymoron
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Let [itex]p_n = 1 + p + p^2 + \dots + p^{n-1}[/itex]. Then we have

[tex]|p_{n+k}-p_n|_p = \left|p^n + p^{n+1} + \dots + p^{n+k-1}\right|_p[/tex]

[tex]= \left|p^n(1+p+p^2 + \dots + p^{k-1})\right|_p[/tex]

[tex]= \frac{1}{p^n}[/tex]

So for any [itex]\epsilon > 0[/itex], we can choose an [itex]N\in\mathbb{N}[/itex] such that [itex]p^N \geq \frac{1}{\epsilon}[/itex], so if [itex]n > N[/itex] we have

[tex]|p_{n+k} - p_n|_p < \frac{1}{p^N} \leq \epsilon[/tex]

Therefore [itex]\{p^n\}_{n\in\mathbb{N}}[/itex] is Cauchy.

- #3

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[tex]|p^n|_p = \frac{1}{p^n} \rightarrow 0 \quad \mbox{as} \quad n\rightarrow \infty[/tex]

the limit

[tex]\lim_{n\rightarrow\infty}^p p^n = 0[/tex]

Hence this sequence is actually a null sequence with respect to the p-adic norm.

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shmoe

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