Detemine whether the interval [4,8] is well-ordered. Explain.

Question

I don't think this interval is well-ordered because the subset (4,8) would not have a smallest value. I'm stuck on how to show (4,8) has no smallest value.

Answer 1

Assume $(4,8)$ has a smallest value. Name it $x$. Now $4<\frac{4+x}{2}:=y<x$. Therefore $x$ is not the smallest value as $y<x$ and $y\in(4,8)$.