Proving Subsets of Intervals in $\mathbb{R}$

[NoName3]

Let $I \subseteq \mathbb{R}$ be an interval. Prove that

1. If $x, y \in I$ and $ x \le y$ then $[x,y] \subseteq I$.

2. If $I$ is an open interval, and if $x \in I$, then there is some $\delta > 0 $ such that $[x-\delta, x+\delta] \subseteq I$.

 

[Greg]

Hello and welcome to MHB, NoName! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?

 

[NoName3]

Hello greg1313, and thanks for the warm welcome. So far I've figured that I'm probably supposed to use part (3) of http://mathhelpboards.com/analysis-50/using-axioms-ordered-field-18031.html#post82951 exercise.

If I understand correctly, that exercise says given any two real numbers $a, b$ such that $a < b$ we can always find a third $c$ such that $a < c < b$ and in particular $c = \frac{1}{2}(a+b)$, but I'm unable to prove it. I'll post an update if I make a progress/figure it out however.