Talk:Closure of Real Interval is Closed Real Interval

Nitpicking/comments:

The proof currently assumes $a\neq b$, but this is not stated as a hypothesis (it doesn't need to be).

The $N_\epsilon$ that is said to exist by definition in the first half of the proof uses an $\epsilon$ that might be different from the $\epsilon$ said to exist when $x=a$. This seems like it should be something like
 * If $x=a$, then $\exists \epsilon'$ such that $0 < \epsilon' < \epsilon$

and make use of the fact that $N_\epsilon$ is an interval. Alternatively, the statement that $N_\epsilon$ exists should be quantified. Right now it looks like one $N_\epsilon$ is being chosen. Mostly, care needs to be taken to ensure that $(c,d)$ and these epsilons all mesh.

In the last part of the proof, when $x<a$, a particular value of $\epsilon$ can be easily selected if you want (say, $(x+a)/2$. I don't know approach is pedagogically superior, or if anyone even has a preference.

Qedetc 01:24, 6 June 2011 (CDT)


 * Good call. Feel free to follow this up. --prime mover 01:35, 6 June 2011 (CDT)

I got carried away and changed it substantially. Let me know if it's too cluttered because of the cases or anything. Qedetc 19:54, 6 June 2011 (CDT)


 * There's still one or two unproved statements. I'll take a good look in due course. --prime mover 01:16, 7 June 2011 (CDT)


 * ... although one thought I'll make: if modifying a proof into a radically different form, it might be better to add it as a different proof. Not sure whether this applies here, as I haven't traced it through to see whether it actually is "the same" or not. --prime mover 01:18, 7 June 2011 (CDT)

I'd say it's essentially the same proof. The idea in both cases is to pick a subinterval $(x,a+\epsilon)$ of $(c,d)$ whose right endpoint is between $a$ and $b$. The $\epsilon$ that was being talked about before is essentially $\min \{d, b\} - a$ in my proof. When I went to try to clarify the epsilon stuff, I just noticed that everything that was going on could be stated in terms of the order and density of the reals. Really, I guess all I did was pick a specific $\epsilon$ and justify its existence.

Qedetc 01:36, 7 June 2011 (CDT)