Recent content by arturo_026

When f maps E into a metric space Y: (E is subset of metric space X) Is it eqivalent to say that f is a continuous mapping and that for a subset E of X, to say that for every p element of E, f is continuous at p.? thank you

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

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.

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...

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...

Hello, the following is a post that was in progress and I am continuing it here after I received a message saying that most of the members had moved from mathhelpforum here. Me: I have a problem where I am asked to show that for a complete metric space X, the the natural Isometry F:X --> X* is...

Yes, now it's clear. Thank you so much Dick, I think I got it!

ok, so now if I replace ≤ by < then I'm able to include all n greater then N and abs. value (s_n) thus becomes only > ε. And this is it?

Thank you for your patience Dick. So I start with this for "s_n converges to zero": For all ε>0, there exists a natural number N such that n≥N implies that abs. value of (s_n - 0) < ε. Now I tried negating every part but it doesn't seem to be right. What seems somewhat correct is that...

Ohhh! in the p-adic metric as n --> inf p^(n+1) goes to zero since p-adic abs. value of p^(n+1) = p^-(n+1)

I see that makes sense. Thank you Dick Now, as far as my semi-complete proof goes, is it correct? and how could I implement the fact that s_n is cauchy in the proof? Thank you again

Thank you for your response Dick. So to prove it is Cauchy I've introduced another sequence that has a summation upper limit of m, so call it s_m. And without loss of generality let n>m. now Apropriately choosing e(epsilon) > p^(-m-1) the for m>N, I can have the p-adic absolute value of (s_n -...

I have the following exercise: Let s_n be a cauchy sequence in Q(rationals) not converging to 0. Show that there exists an e(epsilon) >0 and a natural number N such that either for all n>N, s_n > e or for all n>N, -s_n >e. I know that since Q is not complete, we cannot assume that there...

something I just thought of is that maybe I can apply a version of the ratio test with the p-adic absolute value in place of the normal absolute value? So that way the p-adic abs. value of (p^k+1)/(p^k) = p-adic abs. of p which equals to 1/p [by definition of p-adic abs. value]. So this being...

This is the problem I'm trying to slove: Consider the sequence s_n = Sumation (from k=0 to n) p^k (i.e. s_n=p^0+p^1+p^2...+p^n) in Q(rationals) with the p-adic metric (p is prime). Show that s_n converges to a rational number.[/B] Now, I do get some intuition on showing that the...