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Thread: Cauchy sequences

  1. #1
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    Cauchy sequences

    Assume the sequences $\displaystyle (a_n)$ and $\displaystyle (b_n)$ are Cauchy sequences. Then, by using a triangle inequality argument, prove $\displaystyle c_n = |a_n - b_n|$ is Cauchy.
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  2. #2
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    The key to this proof is the following inequality;
    $\displaystyle \left| {\left| x \right| - \left| y \right|} \right| \le \left| {x - y} \right|$
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  3. #3
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    Quote Originally Posted by Plato View Post
    The key to this proof is the following inequality;
    $\displaystyle \left| {\left| x \right| - \left| y \right|} \right| \le \left| {x - y} \right|$
    How do you incorporate deltas/epsilons in here? Meh confusing proof I suppose.
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  4. #4
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    $\displaystyle \begin{array}{rcl}
    \left| {c_N - c_M } \right| & = & \left| {\left| {a_N - b_N } \right| - \left| {a_M - b_M } \right|} \right| \\
    & \le & \left| {\left( {a_N - b_N } \right) - \left( {a_M - b_M } \right)} \right| \\
    & \le & \left| {\left( {a_N - a_M } \right)} \right| + \left| {\left( {b_N - b_M } \right)} \right| \\
    \end{array}$
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