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Thread: Convergence

  1. #1
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    Convergence

    Hello all,

    I have the following problem:

    Let z_{n} be a complex converging sequence.
    We want to show that \mid z_{n} \mid also converges.

    I have done the following:

    We want to show that \mid z_{n} \mid converges which means that we have to show that for all \epsilon >0 there exists N\leq n such that \vert \vert z_{n}\vert-L \vert <\epsilon - We have:

    \vert \vert z_{n}\vert-L \vert = \vert \vert z_{n}\vert- \vert L \vert \leq  \vert z_{n} -L \vert < \epsilon

    The last inequality comes from the fact that z_{n} converges.

    Is the above proof correct? I think that I am assuming that the limit of \mid z_{n} \mid and z_{n} are the same??!!


    thanks
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  2. #2
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    Re: Convergence

    $L < 0 \Rightarrow ||z_n| - L| \neq ||z_n| - |L||$
    Thanks from surjective
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  3. #3
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    Re: Convergence

    How about rather than "assuming that the limit of z_n and |z_n| and are the same", taking the limit of z_n, which you know exists, and the limit of |z_n| to be |L|.
    Thanks from surjective
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  4. #4
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    Re: Convergence

    Quote Originally Posted by surjective View Post
    I have the following problem:
    Let z_{n} be a complex converging sequence.
    We want to show that \mid z_{n} \mid also converges.
    Actually the problem should be: If $z_n\to L$ then $|z_n|\to |L|$.

    lemma: $||z|-|w||\le|z-w|$
    proof:
    $ \begin{align*}|z|&=|z-w+w|\\&\le|z-w|+|w|\\|z|-|w|&\le|z-w|\\\\|w|&=|w-z+z|\\&\le|w-z|+|w|\\|w|-|z|&\le|w-z| \end{align*}$


    Because $|z-w|=|w-z|$
    $ \begin{align*}-(|z-w|)&\le (|z|-|w|)\le |z-w|\\||z|-|w||&\le |z-w|\end{align*}$
    Thanks from surjective
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