# Thread: coefficient of supremum proof

1. ## coefficient of supremum proof

S is a bounded non empty group.i need to prove that $\displaystyle sup(cS)=csupS$
if c=0 then it trivial
if c>0 then we need to prove that the smallest upper bound to cS is
$\displaystyle csupS$
this is how the book proves:
we take $\displaystyle s\in S$ s<csupS $\displaystyle \frac{s}{c}<supS$ so $\displaystyle \frac{s}{c}$ cannot
be the supremum and there is $\displaystyle x\in S$ for which $\displaystyle x>\frac{s}{c}$.
i cant understand why it proves the law??

2. ## Re: coefficient of supremum proof

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.