Equivalent norms on a field differ only by a power

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I am trying to understand a proof of the following theorem (which i encountered in a P-adic paper i am reading; http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.461.4588&rep=rep1&type=pdf p.20 in the pdf):




Let $||cdot||_1$ and $||cdot||_2$ be two norms on a field F. Then
$||cdot ||_2 sim||cdot||_1$ if and only if there exists a positive real number
$alpha$ such that $||x||_2=||x||_1^alpha , forall xin F$.




There is a proof of the right implication of this theorem in the end of the paper (Picture:)



enter image description here



Unlike the answer to Understanding a statement about equivalent norms ($||cdot ||_2 sim||cdot||_1 $) , this proof seems to be independent of first defining the concept of a topology. The only problem is, I don't understand it... Can anyone please explain the steps of this proof very carefully?



I follow the proof until the part where we prove that $||x^m/a^n||_2 $ approaches zero. How do we do that? Also, how can i argue that it is possible to choose sequences $m_k$ and $n_k$ such that $||x^m_k/a^n_k ||_1$ approaches 1 as k increases in the first place? Isn't this dependent on the norm in question at all?



Thanks :)




metric-spaces cauchy-sequences p-adic-number-theory




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    up vote
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    down vote

    favorite












    I am trying to understand a proof of the following theorem (which i encountered in a P-adic paper i am reading; http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.461.4588&rep=rep1&type=pdf p.20 in the pdf):




    Let $||cdot||_1$ and $||cdot||_2$ be two norms on a field F. Then
    $||cdot ||_2 sim||cdot||_1$ if and only if there exists a positive real number
    $alpha$ such that $||x||_2=||x||_1^alpha , forall xin F$.




    There is a proof of the right implication of this theorem in the end of the paper (Picture:)



    enter image description here



    Unlike the answer to Understanding a statement about equivalent norms ($||cdot ||_2 sim||cdot||_1 $) , this proof seems to be independent of first defining the concept of a topology. The only problem is, I don't understand it... Can anyone please explain the steps of this proof very carefully?



    I follow the proof until the part where we prove that $||x^m/a^n||_2 $ approaches zero. How do we do that? Also, how can i argue that it is possible to choose sequences $m_k$ and $n_k$ such that $||x^m_k/a^n_k ||_1$ approaches 1 as k increases in the first place? Isn't this dependent on the norm in question at all?



    Thanks :)




    metric-spaces cauchy-sequences p-adic-number-theory




    share|cite|improve this question





















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      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I am trying to understand a proof of the following theorem (which i encountered in a P-adic paper i am reading; http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.461.4588&rep=rep1&type=pdf p.20 in the pdf):




      Let $||cdot||_1$ and $||cdot||_2$ be two norms on a field F. Then
      $||cdot ||_2 sim||cdot||_1$ if and only if there exists a positive real number
      $alpha$ such that $||x||_2=||x||_1^alpha , forall xin F$.




      There is a proof of the right implication of this theorem in the end of the paper (Picture:)



      enter image description here



      Unlike the answer to Understanding a statement about equivalent norms ($||cdot ||_2 sim||cdot||_1 $) , this proof seems to be independent of first defining the concept of a topology. The only problem is, I don't understand it... Can anyone please explain the steps of this proof very carefully?



      I follow the proof until the part where we prove that $||x^m/a^n||_2 $ approaches zero. How do we do that? Also, how can i argue that it is possible to choose sequences $m_k$ and $n_k$ such that $||x^m_k/a^n_k ||_1$ approaches 1 as k increases in the first place? Isn't this dependent on the norm in question at all?



      Thanks :)




      metric-spaces cauchy-sequences p-adic-number-theory




      share|cite|improve this question











      I am trying to understand a proof of the following theorem (which i encountered in a P-adic paper i am reading; http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.461.4588&rep=rep1&type=pdf p.20 in the pdf):




      Let $||cdot||_1$ and $||cdot||_2$ be two norms on a field F. Then
      $||cdot ||_2 sim||cdot||_1$ if and only if there exists a positive real number
      $alpha$ such that $||x||_2=||x||_1^alpha , forall xin F$.




      There is a proof of the right implication of this theorem in the end of the paper (Picture:)



      enter image description here



      Unlike the answer to Understanding a statement about equivalent norms ($||cdot ||_2 sim||cdot||_1 $) , this proof seems to be independent of first defining the concept of a topology. The only problem is, I don't understand it... Can anyone please explain the steps of this proof very carefully?



      I follow the proof until the part where we prove that $||x^m/a^n||_2 $ approaches zero. How do we do that? Also, how can i argue that it is possible to choose sequences $m_k$ and $n_k$ such that $||x^m_k/a^n_k ||_1$ approaches 1 as k increases in the first place? Isn't this dependent on the norm in question at all?



      Thanks :)





      metric-spaces cauchy-sequences p-adic-number-theory





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