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

In summary, The conversation is about solving a question related to the p-adic metric and cauchy sequences. The question is to show that the sequence s_n is cauchy, but cannot be expressed by a rational number. The participants discuss using the p-adic absolute value and the cauchy criterion to prove this. They also mention that a series converges in Q_p if and only if the terms approach 0 in the p-adic metric.
  • #1
arturo_026
18
0
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.
 
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  • #2
Look at |s_n - s_m|. What power of p will divide this?
 
  • #3
morphism said:
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.
 
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  • #4
I think you have the right idea.
 
  • #5
morphism said:
I think you have the right idea.

Great! thank you very much. I'll work on cleaning it up.
 
  • #6
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.
 

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

What is a p-adic metric?

A p-adic metric is a mathematical concept used to measure the distance between numbers in a specific way. It is based on the idea of dividing numbers into smaller and smaller parts, rather than just subtracting them from one another.

What does it mean for a sequence to be cauchy?

A sequence is considered cauchy if the terms in the sequence get closer and closer together as the sequence progresses. In other words, for any small distance, there is a point in the sequence where all following terms are within that distance from each other.

How is a sequence in Q with p-adic metric different from a sequence in Q with the standard metric?

The p-adic metric and the standard metric both measure the distance between numbers, but they do so in different ways. The p-adic metric takes into account the prime factorization of numbers, while the standard metric uses the absolute value difference between numbers.

What are some examples of sequences in Q with p-adic metric that are cauchy?

One example of such a sequence is 1, 1/2, 1/4, 1/8, 1/16, ... where the terms get closer and closer to 0 as the sequence progresses. Another example is 1, -1/2, 1/3, -1/4, 1/5, -1/6, ... where the terms alternate between positive and negative, but still get closer to 0.

Why is the concept of cauchy sequences in Q with p-adic metric important in mathematics?

The concept of cauchy sequences with p-adic metric is important because it is used in various branches of mathematics, such as number theory and algebraic geometry. It also has applications in cryptography and coding theory. Additionally, understanding cauchy sequences helps to better understand the properties and structure of the rational numbers.

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