Trait proof with absolute value Will be happy if you guys check my proof.
I have to prove that:
$$ |x| < m \iff -m < x < m$$
so I'm going back to the basic definition of the absolute value:
$$ x < 0 \implies |x| = -x$$
$$ x > 0 \implies |x| = x$$
considering $x$ in both cases and createing to inequalities:
$$-x < m \implies x > -m $$ 
$$ x < m$$
now from the two inequalities above, concluding about the range of $x$  :
$$-m<x<m$$
thanks in advance :)

Your proof is essentially correct. I would suggest just a little adjustment.
1) For $x<0$ you got $x>-m$ so you have $-m<x<0$;
2) For $x\ge 0$ you got $x<m$ so you have $0\le x<m$;
Putting both together you get:
$$-m<x<m$$
