actually I could shorten it even more and say because x in B then by definition of y=sup(B) for every x in B y >or= x.
But then again I'm not sure I can say x is in B.
I'm supposed to prove that if A is a subset of B then sup(A) <or= to sup(B), provided that these least upper bounds exist.
So I started by using the definition of a least upper bound. I let x = sup(A) therefore by definition there exists a x in B such that for ever a in A, a <or= to x.
also there exists a c in B such that for every a in A, a <or= c and c >or= x.
My problem is though I'm not completely sure though whether c is necessarily in B. because then I let y=sup(B) and say because c in B, by definition y >or= c.
Then by transitivity x <or= y.
Also if there are any ideas of how this proof could be done better I'd appreciate it because this doesnt seem solid enough.