Suppose $S \subset \mathbb{R}$ is bounded above but has no maximum element. Prove that sup $S \notin S$.

Question

I've been trying to prove this and I just can't seem to figure it out. I'm just totally lost as to where to even start. I tried applying a method that was shown here but I know that $S$ does not have to necessarily be of the form $[a,b)$ (or $(a,b)$, etc.)

So I'm pretty much stuck, I'd really appreciate some help, or someone to point me in the right direction.

Thanks in advance.

Prove that if $\sup S$ were in $S$, it would be the maximum element of $S$. — Brian M. Scott, Mar 16 '21 at 21:34

score 0 · Accepted Answer · answered Mar 16 '21 at 21:41

0

Just approach the problem via contradiction. Let $sup S=d$. Then $d\geq s$ (by definition of supremum) for every $s\in S$. Now, if $d\in S$, then it is trivial to see that $d$ is a maximum of $S$. However, this is a contradiction to the fact that $S$ has no maximum.

Consequently $d\notin S$.

Actually, supremum of a set is called maximum if it is one of the member of the set.

answered Mar 16 '21 at 21:41

pmun

1,460

Oh my gosh, I don't know how I didn't even think of that. I really need to get a better hang of this if I want to pursue my undergrad degree in mathematics. – reyspawne Mar 16 '21 at 23:48

Suppose $S \subset \mathbb{R}$ is bounded above but has no maximum element. Prove that sup $S \notin S$.

1 Answers1