Prove S = sup A if and only if S is upper bound of A and

[transgalactic]

A is a non empty group which is bounded from the top. prove that S=sup A if and only if S is the upper bound of of A and there i a series of numbers An (An which belongs to A) so An->S

i was proposed a general solution: presume that supA=S show that there is a series An (An belongs to A) so lim An=s (n goes to infinity)

and there is a sketch here:
http://img66.imageshack.us/img66/3286/img9144rf7.jpg

how to make a formal proof for it??