Prove that the set (-2, ∞) is unbounded above and has no minimum element.
Help please
Suppose that $\displaystyle \mathcal{S}=(-2,\infty)$.
unbounded above. If $\displaystyle t\in\mathcal{S}$ we know that $\displaystyle 0<1$ so $\displaystyle t<t+1$.
has no minimum element.. If $\displaystyle t\in\mathcal{S}$ then $\displaystyle -2<t$.
So $\displaystyle -2<\frac{t+2}{2}<t$. Can $\displaystyle t$ be minimal in $\displaystyle \mathcal{S}~?$
S has no upper bound
Proof by contradiction: Suppose S has an upperbound, a. Certainly, 0 is in S so we must have 0< a. Then a< a+ 1 so a+ 1 is not in S. But S contains all numbers larger than -2 so that is a contradiction
S has no smallest member.
Proof by contradicton: Suppose S has smallest member b. Since b is in S, -2< b. Now, -2< (b- 2)/2< b is larger than -2 so in S but less than b which contradicts the assumption that b is the smallest member of S.