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I've been studying p-adic analysis recently and after a bit of searching on the web, I haven't found an answer as to why the Euclidean metric is referred to as the 'prime at infinity', and given the suggestive notation of $|\cdot |_{\infty}$. Does the p-adic metric for larger and larger primes serve as a better and better approximation of the Euclidean metric, or is it something else?

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This article looks like it might elucidate what you're asking. I only skimmed it but it looks pretty thorough? drchristiansalas.wordpress.com/2013/03/02/… –  Ian Coley Mar 20 '13 at 7:05
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See also this question. –  Zhen Lin Mar 20 '13 at 8:07
    
@ZhenLin: The answer in that question is sufficiently far above my head that I can't quite tell whether it would answer my question, but it seems like it might, thanks. –  cactuar Mar 20 '13 at 17:49

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