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Show a certain sequence in Q, with p-adict metric is cauchy


New member
Mar 7, 2012
I left this following question from my excercises last, hoping that solving the others will give me an insight onto how to proceed. But I still don't have a plan on how to start it:

Consider the sequence s_n = Ʃ (k=0 to n) (t_k * p^k) in Q(rationals) with the p-adic metric (p is prime); where t_n is the sequence 1,2,1,1,2,1,1,1,2,... Show that s_n is Cauchy, but [s_n] (the equivalence class of s_n) cannot be expressed by a rational number.

As far as what I know: I'm familiar with the p-adic metric, and what a cauchy sequence is. I just cant think of how to show that s_n is cauchy.

Thank you very much for any advice and hints.


New member
Mar 3, 2012
Well, when in doubt, just start with the definition and see if you can shove your particular case into it. Can you bound the difference between the j-th and k-th terms, for j<k, in terms of j? Maybe by considering the difference between the j-th and (j+1)-st terms, and summing them up, like you do to show the series 2^-n is Cauchy?