# Show that the ﬁeld of p-adic numbers is complete

this is a question from a book I'm struggling with, please could you provide a clear proof

Show that the ﬁeld of p-adic numbers is complete

i.e. that a sequence of p-adic numbers converges if and only if it is Cauchy?

kind thanks

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More often than not, I have seen the $p$-adic numbers defined as the metric completion of $\Bbb Q$ under the $p$-norm. By this definition it is trivial that the $p$-adic numbers are complete. So I suppose this is not the definition you are using. How did you define them? –  Asaf Karagila Mar 30 '13 at 11:41
@AsafKaragila : not entirely trivial. The completion $\overline X$ of a metric space $X$ is obtained "adding" the limits of the Cauchy sequences in $X$. Then you have to say something about the Cauchy sequences in $\overline X$ before you can conclude that $\overline X$ is itself complete. Yet, I agree that the result is part of the construction. –  Andrea Mori Mar 30 '13 at 12:23
@Andrea: When defining "the completion" one has to show two things, (1) it's unique up to an isometry; (2) it's complete. Otherwise the name "the completion" is misleading. I assume that when defining the $p$-adic numbers as the metric completion of something, one already knows that the completion metric space is a uniquely generated complete metric space. –  Asaf Karagila Mar 30 '13 at 12:27
@AsafKaragila : of course you are right about the terminology of "completion" already including the basic properties. Given the way the question is posed, I doubt that the questioner is thinking about the $p$-adic numbers as the completion of $\Bbb Q$ under the $p$-adic norm and my comment had the purpose to clarify (for him) the situation. –  Andrea Mori Mar 30 '13 at 12:44
@Andrea Mori so I guess the only thing that needs to be shown here is that that a sequence of p-adic numbers converges if and only if it is Cauchy. so this is different to the standard analysis, in which cauchy does not necessarily imply conv. could you give me some advice how to go about this. What exactly is a p-adic number? I understand all the details about p adic order and rules for p adic convergence of seq. and series. kind thanks for pointing me in right direction –  Mathproof P. Apr 6 '13 at 22:34