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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 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 $$c \lt b$$, then we can pick $$d \in ( c, c + \epsilon )$$ such that $$c \lt d \lt b$$ ... ... "My question is as follows:
How (... rigorously speaking ... ) do we know such a $$d$$ exists ...
In other words, what is the rigorous justification that if $$c \lt b$$, the we can pick $$d \in ( c, c + \epsilon )$$ such that $$c \lt d \lt b$$ ... ...?
Question 2
In the above proof by Sohrab we read the following:
" ... ... Now, if $$c \lt b$$, then we can pick $$d \in ( c, c + \epsilon )$$ such that $$c \lt d \lt b$$, and it follows that $$[a,d ]$$ can also be covered by a finite subcover. i.e. $$d \in S$$ ... ... "Can someone please explain why/how it follows that $$[a,d ]$$ can also be covered by a finite subcover. i.e. $$d \in S$$ ... ... ?
Help will be appreciated ...
Peter
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 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 $$c \lt b$$, then we can pick $$d \in ( c, c + \epsilon )$$ such that $$c \lt d \lt b$$ ... ... "My question is as follows:
How (... rigorously speaking ... ) do we know such a $$d$$ exists ...
In other words, what is the rigorous justification that if $$c \lt b$$, the we can pick $$d \in ( c, c + \epsilon )$$ such that $$c \lt d \lt b$$ ... ...?
Question 2
In the above proof by Sohrab we read the following:
" ... ... Now, if $$c \lt b$$, then we can pick $$d \in ( c, c + \epsilon )$$ such that $$c \lt d \lt b$$, and it follows that $$[a,d ]$$ can also be covered by a finite subcover. i.e. $$d \in S$$ ... ... "Can someone please explain why/how it follows that $$[a,d ]$$ can also be covered by a finite subcover. i.e. $$d \in S$$ ... ... ?
Help will be appreciated ...
Peter
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