The question is:
assume A is a proper subset of B and both belongs to R^n dimention
show that this implies sup(A)< and= sup(B).
It looks quite obvious, but I don't know how to construct the proof formally.
Thanks for help.
Actually, you have a serious problem right from the start. While there is a standard "order" on R, there is no standard order on R^n for n> 1. And without a specific order there is no "lim" or "sup".
The statement you give is easy to prove in R but makes no sense in R^n for n> 1.