What is the relationship between sup of unbounded sets in real numbers?

[v.rad]

1. sup (empty set) = -infinity, and if V is not bounded above, then sup V = +infinity. Prove if V$$\subseteq$$W$$\subseteq$$Real Numbers then sup V is lessthan/equalto supW




3. I used a proof by contrapositive, but I'm not sure if it is completely valid...

 

[hunt_mat]

I am assuming that these are intevals. You know that the lemma is clear if V=W, don't you? So assume that
[tex]
V\subset W
[\tex]
So there is an element w in W which is not an element of V, examine |w-sup(V)|.

 

[HallsofIvy]

No reason to assume these are intervals.

Yes, the contrapositive is the way to go. Suppose sup(W)> sup(V). Then there exist x such that sup(V)< x< sup(W). From that it follows that there exist a member of W larger than x and so larger than any member of V, a contradiction.