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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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