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Proof: c * divergent sequence diverges

  1. Oct 13, 2009 #1
    1. The problem statement, all variables and given/known data

    Suppose that {a_n} is a divergent sequence of real numbers and c \in R, c <> 0. Prove that {c*a_n} diverges.

    2. Relevant equations



    3. The attempt at a solution I have attempted to solve the problem as a proof by contradiction, but am afraid I am leaving something out. Please confirm my proof is complete or prompt me to add. Thanks!

    Proof is by contradiction. Suppose {c*a_n} is convergent. This means that |c*a_n - L| < e, for e > 0. Then there is N \in N such that |a_n - L/c| < e/|c|, n >= N. But this is the definition of limit of a sequence, so a_n converges. But this contradicts our problem statement so, in fact, a_n diverges. End of proof.
     
  2. jcsd
  3. Oct 13, 2009 #2

    LCKurtz

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    You have the right idea, but it isn't written well. You need to use your observations to write your argument in reverse. Something like this:

    Given can [itex]\rightarrow[/itex] L we will show that an [itex]\rightarrow[/itex] L/|c|, which is a contradiction.

    Suppose[itex]\ \epsilon > 0[/itex]. Then there is N > 0 such that:

    | can - L| < |c|[itex]\epsilon [/itex]

    for all n > N. Therefore

    |can - L| = |c(an - L/c| = |c||(an - L/c| < |c|[itex]\epsilon[/itex]

    which gives, upon dividing that last inequality by |c|, | an - L/c| < [itex]\epsilon[/itex]
    for all n > N.
     
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