It doesn't. I recommend you go back and re-read your book. You are saying that you want to prove that c sup(S) is equal to sup(cS) and, if it were correct, what you write would prove it is NOT.
S is a bounded non empty group.i need to prove thati cant understand why it proves the law??
if c=0 then it trivial
if c>0 then we need to prove that the smallest upper bound to cS is
this is how the book proves:
we take s<csupS so cannot
be the supremum and there is for which .