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

[arturo_026]

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 can't think of how to show that s_n is cauchy.

Thank you very much for any advice and hints.

 

[morphism]

Look at |s_n - s_m|. What power of p will divide this?

 

[arturo_026]

Look at |s_n - s_m|. What power of p will divide this?

Will it be the p-adic absolute value of the partial series from m+1 to n, so that way if I choose N large enough so p^-N is larger or equal to such series (for any n), and p^-N≥ε , then s_n will satisfy the cauchy criterion.

 

[morphism]

I think you have the right idea.

 

[arturo_026]

I think you have the right idea.

Great! thank you very much. I'll work on cleaning it up.

 

[morphism]

No problem. By the way, in general, a series $\sum a_n$ converges in $\mathbb Q_p$ iff $a_n \to 0$ in the p-adic metric (compare to the case in $\mathbb R$ or $\mathbb C$!). The proof of this general statement should be similar to the proof you're writing up.