if f is bounded in a non empty group D and if c<0 then cf is bounded and

*-multiply

cf- c*f

then prove that:

sup(cf)(D)=c*inf(f)(D)

inf(cf)(D)=c*inf(f)(D)

the prove in the book says

L<=f<=M

cL>=cf>=cM

so cL is upper bound cM is lower bound

then they say cf(D)=(cf)(D)

i know that cf(D) is multiplication of every member by c in the group

what is the meaning of (cf)(D)??

then they say that

c*supA=infc*A

for c<0

i dont know wht they suddenly say that as if it was given

i dont know how they showed that

c*supA=infc*A

for c<0