S is a bounded non empty group.i need to prove that $\displaystyle sup(cS)=csupS$i 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

$\displaystyle csupS$

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 $\displaystyle x>\frac{s}{c}$.