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Math Help - Cauchy sequences

  1. #1
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    Cauchy sequences

    Assume the sequences (a_n) and (b_n) are Cauchy sequences. Then, by using a triangle inequality argument, prove 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;
    \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;
    \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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    \begin{array}{rcl}<br />
  \left| {c_N  - c_M } \right| & = & \left| {\left| {a_N  - b_N } \right| - \left| {a_M  - b_M } \right|} \right| \\ <br />
  & \le & \left| {\left( {a_N  - b_N } \right) - \left( {a_M  - b_M } \right)} \right| \\ <br />
  & \le & \left| {\left( {a_N  - a_M } \right)} \right| + \left| {\left( {b_N  - b_M } \right)} \right| \\ <br />
 \end{array}
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