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Show that $\mathbb{Z}_p$ = $\varprojlim_n \mathbb{Z}/p^n$ is the completion of $\mathbb{Z}$ with respect to the metric $(x, y) \rightarrow \|x-y\|_p$, i.e, the p-adic metric.

I've tried doing this with cauchy sequences, but I don't think it's right. I feel like this is pretty standard. Is there a way to do it with imbeddings?

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You need to show that $\mathbb{Z}_p$ satisfies the universal property of completions: If $C$ is any complete metric space, and $f:\mathbb{Z} \to C$ is an embedding, then $f$ extends uniquely to an embedding from $\mathbb{Z}_p \to C$. (Here, "embedding" means "continuous, distance-preserving embedding".)

Use the fact that every element of $\mathbb{Z}_p$ is the limit of a Cauchy sequence of elements from $\mathbb{Z}$.