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

EberthQ

New member
Mar 7, 2012
3
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.
 

Tinyboss

New member
Mar 3, 2012
7
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?