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