I'm asked the following:
Prove that if b<0, then inf(bS)=bsup(S).
My attempt was as follows:
Proof:
Let b<0 and sup(S)=X
This gives that X$\leq$s for s$\epsilon$S.
This gives that bX is the greatest lower bound for bS. *****
Therefore bX=inf(bS) (1)
Now
Let b<0 and inf(bS)=Y
For all z$\epsilon$S, we have Y$\leq$bz, equivalent to (y/b)$\leq$z
This gives that y/b is an upper bound for S.
For all z$\epsilon$S, Y/b=sup(S)
This gives y=bsup(S) (2)
Combining (1) and (2) completes the proof.
QED
&&&&&
I turned this in a little while ago and received it back having only received half credit. Professor says I should know my mistake but I can't identify what it is. He marked the lines around the *****.
Please provide corrections to my proof.
Additionally, no mater the question I ask, I get downvoted and I'm not sure what I am doing wrong when I ask my questions. Advise to prevent this is also appreciated, thanks.