Because a divides b we have that there is an integer c such that b=ca. Similarly because b divides a we have that there is another integer d such that a=db. But then putting everything together we get that b=ca=cdb and by canceling b from both sides (b is non zero) we find that 1=cd. But since c and d are integers and both of them divide 1 and since 1 has only two divisors among the integer family (namely 1 and -1 are the only divisors of number 1) of numbers we obtain that either c=1 or c=-1. So either a=b or a=-b. This finishes the proof.
