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

In summary, the conversation discusses the proof of two statements. The first statement states that if $x$ and $y$ are elements of an interval $I$ and $x$ is less than or equal to $y$, then the closed interval $[x,y]$ is a subset of $I$. The second statement states that if $I$ is an open interval and $x$ is an element of $I$, then there exists a positive number $\delta$ such that the closed interval $[x-\delta, x+\delta]$ is a subset of $I$. The conversation also mentions using part (3) of a previous exercise to prove these statements, but the individual is currently unable to do so.
  • #1
NoName3
25
0
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 $[x-\delta, x+\delta] \subseteq I$.
 
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  • #2
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?
 
  • #3
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.
 

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

What is a subset of an interval in $\mathbb{R}$?

A subset of an interval in $\mathbb{R}$ is a set of real numbers that is contained within the boundaries of the interval. For example, the subset [2,5] would include all real numbers between 2 and 5, including 2 and 5 themselves.

How is a subset of an interval in $\mathbb{R}$ proven?

A subset of an interval in $\mathbb{R}$ can be proven by showing that all elements of the subset are also elements of the interval. This can be done by using the definition of an interval and checking each element against the boundaries of the interval.

What is the notation for a subset of an interval in $\mathbb{R}$?

The notation for a subset of an interval in $\mathbb{R}$ is a bracket notation, such as [a,b] or (a,b). The square brackets indicate that the endpoints are included in the subset, while the parentheses indicate that the endpoints are not included.

Can a subset of an interval in $\mathbb{R}$ be an empty set?

Yes, a subset of an interval in $\mathbb{R}$ can be an empty set. This would occur if there are no real numbers between the boundaries of the interval. For example, the subset (5,5) would be an empty set since there are no real numbers between 5 and 5.

What is the importance of proving subsets of intervals in $\mathbb{R}$?

Proving subsets of intervals in $\mathbb{R}$ is important in mathematics because it allows us to make accurate statements about the relationship between sets of real numbers. It also helps us to better understand the properties of intervals and how they can be used in various mathematical concepts and applications.

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