This is a well known inequality.
From that the proof is trivial.
Prove that if an converges to A, then |an| converges to |A|. Then prove or disprove the converse to this.
if an converges to A, then |an-A|<epsilon for n>=N. So do we just approach this first for an positive and then negative to show it converges to |A|. I don't know how to show this formally... I would say the converse is likely false since it could be negative.