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.

 

[Tinyboss]

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?

 

[math771]

Hey there! I can definitely understand your struggle with this exercise. It seems like you have a good understanding of the concepts involved, but just need some guidance on how to approach the problem.

First, let's review what it means for a sequence to be Cauchy. A sequence is Cauchy if for any positive real number ε, there exists a natural number N such that for all n,m ≥ N, |s_n - s_m| < ε. In other words, as n and m get larger, the terms in the sequence get closer and closer together.

Now, let's think about how we can show that s_n is Cauchy in this particular case. One approach could be to use the fact that t_n is a sequence of 1s and 2s, and see how that affects the terms in s_n. Another approach could be to use the definition of the p-adic metric and see how it relates to the terms in s_n.

As for the second part of the exercise, it's asking you to show that [s_n] cannot be expressed by a rational number. This means that there is no rational number that is equivalent to the sequence s_n. Again, you can use the definitions and properties of the p-adic metric to help you with this.

I hope this gives you some ideas on how to start the exercise. Don't be afraid to try out different approaches and see what works. Good luck!