# Supremums of unbounded sets

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
Homework Helper
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