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I am reading Tom Apostol's book: Mathematical Analysis (Second Edition).
I am currently studying Chapter 4: Limits and Continuity.
I am having trouble in fully understanding the proof of Bolzano's Theorem (Apostol Theorem 4.32).
Bolzano's Theorem and its proof reads as follows:
https://www.physicsforums.com/attachments/3863
In the above proof, Apostol writes the following:" ... ... If \(\displaystyle f(c) \lt 0\), then \(\displaystyle c - \delta/2\) is an upper bound for A, again contradicting the definition of \(\displaystyle c\). ... ... "
Can someone please explain why \(\displaystyle f(c) \lt 0\) implies that \(\displaystyle c - \delta/2\) is an upper bound for A?
Help will be appreciated ... ...
Peter
I am currently studying Chapter 4: Limits and Continuity.
I am having trouble in fully understanding the proof of Bolzano's Theorem (Apostol Theorem 4.32).
Bolzano's Theorem and its proof reads as follows:
https://www.physicsforums.com/attachments/3863
In the above proof, Apostol writes the following:" ... ... If \(\displaystyle f(c) \lt 0\), then \(\displaystyle c - \delta/2\) is an upper bound for A, again contradicting the definition of \(\displaystyle c\). ... ... "
Can someone please explain why \(\displaystyle f(c) \lt 0\) implies that \(\displaystyle c - \delta/2\) is an upper bound for A?
Help will be appreciated ... ...
Peter
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