# Thread: Cauchy sequences

1. ## 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.

2. The key to this proof is the following inequality;
$\displaystyle \left| {\left| x \right| - \left| y \right|} \right| \le \left| {x - y} \right|$

3. Originally Posted by Plato
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.

4. $\displaystyle \begin{array}{rcl} \left| {c_N - c_M } \right| & = & \left| {\left| {a_N - b_N } \right| - \left| {a_M - b_M } \right|} \right| \\ & \le & \left| {\left( {a_N - b_N } \right) - \left( {a_M - b_M } \right)} \right| \\ & \le & \left| {\left( {a_N - a_M } \right)} \right| + \left| {\left( {b_N - b_M } \right)} \right| \\ \end{array}$