Here is the problem:
you now that both $\displaystyle (a_n)$ and $\displaystyle (b_n)$ are Cauchy, so given that $\displaystyle m,n>N\in\mathbb{N}$ and $\displaystyle \epsilon >0$ we have that
$\displaystyle \|a_n-a_m\|<\frac{\epsilon}{2}$ and
$\displaystyle \|b_n-b_m\|<\frac{\epsilon}{2}$ whenever $\displaystyle m,n>N$
now you need to look at
$\displaystyle \|c_n-c_m\|=\| |a_n-b_n|-|a_m-b_m|\|$
and you need to show that (using the triangle inequality)
$\displaystyle \|c_n-c_m\|<\|a_n-a_m\|+\|b_n-b_m\|<\epsilon$
to show that $\displaystyle (c_m)$ is Cauchy
$\displaystyle |c_{n}-c_{m}| =||a_{n}-b_{n}|-|a_{m}-b_{m}||\leq |(a_{n}-b_{n})-(a_{m}-b_{m})|=$$\displaystyle |(a_{n}-a_{m})+(b_{m}-b_{n}|\leq |a_{n}-a_{m}|+|b_{n}-b_{m}|$
And since {$\displaystyle a_{n}$} and {$\displaystyle b_{n}$} are Caychy sequences ,given ε>0 there exist natural Nos $\displaystyle k_{1}$ $\displaystyle k_{2}$ such that:
for n,m$\displaystyle \geq k_{1}$ ,then $\displaystyle |a_{n}-a_{m}|<\frac{\epsilon}{2}$ ,and
for n,m$\displaystyle \geq k_{2}$ ,then $\displaystyle |b_{n}-b{m}|<\frac{\epsilon}{2}$.
Now choose k = max($\displaystyle k_{1}$,$\displaystyle k_{2}$)
e.t.c e.t.c