# Cauchy sequence

Question

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

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Solution

Let $p_n = 1 + p + p^2 + \dots + p^{n-1}$. Then we have

$$|p_{n+k}-p_n|_p = \left|p^n + p^{n+1} + \dots + p^{n+k-1}\right|_p$$
$$= \left|p^n(1+p+p^2 + \dots + p^{k-1})\right|_p$$
$$= \frac{1}{p^n}$$

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

$$|p_{n+k} - p_n|_p < \frac{1}{p^N} \leq \epsilon$$

Therefore $\{p^n\}_{n\in\mathbb{N}}$ is Cauchy.

ii) Since

$$|p^n|_p = \frac{1}{p^n} \rightarrow 0 \quad \mbox{as} \quad n\rightarrow \infty$$

the limit

$$\lim_{n\rightarrow\infty}^p p^n = 0$$

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

shmoe