# Closed and Bounded Intervals are Compact .... Sohrab, Propostion 4.1.9 .... ....

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In summary, Sohrab proves that if c is less than b, then there exists a d such that c<d<b. This means that [a,d] can be covered by a finite subcover.
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Closed and Bounded Intervals are Compact ... Sohrab, Proposition 4.1.9 ... ...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).

I am focused on Chapter 4: Topology of [FONT=MathJax_AMS]R[/FONT] and Continuity ... ...

I need help in order to fully understand the proof of Proposition 4.1.9 ...Proposition 4.1.9 and its proof read as follows:
View attachment 9091
My questions are as follows:Question 1

In the above proof by Sohrab we read the following:

" ... ... Now, if $$\displaystyle c \lt b$$, then we can pick $$\displaystyle d \in ( c, c + \epsilon )$$ such that $$\displaystyle c \lt d \lt b$$ ... ... "My question is as follows:

How (... rigorously speaking ... ) do we know such a $$\displaystyle d$$ exists ...

In other words, what is the rigorous justification that if $$\displaystyle c \lt b$$, the we can pick $$\displaystyle d \in ( c, c + \epsilon )$$ such that $$\displaystyle c \lt d \lt b$$ ... ...?

Question 2

In the above proof by Sohrab we read the following:

" ... ... Now, if $$\displaystyle c \lt b$$, then we can pick $$\displaystyle d \in ( c, c + \epsilon )$$ such that $$\displaystyle c \lt d \lt b$$, and it follows that $$\displaystyle [a,d ]$$ can also be covered by a finite subcover. i.e. $$\displaystyle d \in S$$ ... ... "Can someone please explain why/how it follows that $$\displaystyle [a,d ]$$ can also be covered by a finite subcover. i.e. $$\displaystyle d \in S$$ ... ... ?

Help will be appreciated ...

Peter

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Hi Peter,

For question 1, since $c<b$ and $\epsilon>0$, you can take $d$ between $c$ and $\min(c+\epsilon,b)$.

For question 2, since $c<d<c+\epsilon$, and $(c-\epsilon,c+\epsilon)\subset O_\lambda$, $d\in O_\lambda$, and you can add $O_\lambda$ to the finite cover of $[a,c)$ to get a finite cover of $[a,d]$.

Hi Peter,

My answer of question 2 is not quite correct. In fact, we do not necessarily have a finite cover of the interval $[a,c)$ (which is not compact).

We do not know if $c\in S$. What we do know is that, for any $x$ with $a<x<c$, we have a finite cover $U_x$ of $[a,x]$. We can take $x$ between $c-\epsilon$ and $c$, which gives the inequalities:
$$a<c-\epsilon<x<c<d<c+\epsilon$$
Now, $U_x$ covers $[a,x]$ and $O_\lambda$ covers $(c-\epsilon,c+\epsilon)$ (which contains $d$); therefore $U_x\cup O_\lambda$ covers $[a,d]$; this means that $d\in S$, which contradicts $c=\sup S$.

castor28 said:
Hi Peter,

My answer of question 2 is not quite correct. In fact, we do not necessarily have a finite cover of the interval $[a,c)$ (which is not compact).

We do not know if $c\in S$. What we do know is that, for any $x$ with $a<x<c$, we have a finite cover $U_x$ of $[a,x]$. We can take $x$ between $c-\epsilon$ and $c$, which gives the inequalities:
$$a<c-\epsilon<x<c<d<c+\epsilon$$
Now, $U_x$ covers $[a,x]$ and $O_\lambda$ covers $(c-\epsilon,c+\epsilon)$ (which contains $d$); therefore $U_x\cup O_\lambda$ covers $[a,d]$; this means that $d\in S$, which contradicts $c=\sup S$.

Thanks for the guidance and assistance ... and the correction ...

Peter

## 1. What is the definition of a closed interval?

A closed interval is a set of real numbers that includes both of its endpoints. It is represented by the notation [a, b], where a and b are the endpoints and a ≤ b.

## 2. What is a bounded interval?

A bounded interval is a set of real numbers that has finite upper and lower bounds. It is represented by the notation (a, b), where a and b are the upper and lower bounds, and a < b.

## 3. What does it mean for an interval to be compact?

A compact interval is a set of real numbers that is both closed and bounded. This means that it includes its endpoints and has finite upper and lower bounds.

## 4. What is Proposition 4.1.9 in Sohrab's book?

Proposition 4.1.9 in Sohrab's book states that a closed and bounded interval is compact. This means that if a set of real numbers is both closed and bounded, then it is also compact.

## 5. How is the compactness of closed and bounded intervals useful in mathematics?

The compactness of closed and bounded intervals is useful in many areas of mathematics, such as analysis, topology, and calculus. It allows for the use of important theorems and techniques, such as the Heine-Borel theorem and the Bolzano-Weierstrass theorem, which help to prove the existence and convergence of solutions to problems.

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