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Limit of a sequence does not goes to zero

  1. Nov 1, 2013 #1
    1. The problem statement, all variables and given/known data
    Could you guys please verify this proof of mine? I want to show that a limit with particular property does not go to 0. It is part of the proof that when a sequence have an ever increasing term then the limit of the sequence is not 0.

    3. The attempt at a solution

    The ##\lim_{n \to \infty} a_n \neq 0## if ##|a_n|<|a_{n+1}|##

    Suppose ##\lim_{n \to \infty} a_n = 0##, then there exist ##n>0##, such that ##|a_N| < \epsilon## when ##N>n##. Taking an arbitrary ##a_n## as ##\epsilon##, we can get ##|a_N| < |a_n|.## Because it also true that ##N=n+1>n##, we get ##|a_n|>|a_{n+1}|##. A contradiction.

    Hence it is not possible for the limit to be 0.

    Thank You
     
  2. jcsd
  3. Nov 1, 2013 #2

    Dick

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    Science Advisor
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    That looks just fine to me.
     
  4. Nov 1, 2013 #3
    Ok thanks for your verification Dick! I'm quite happy to discover this by myself.
     
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