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I am reading "Introduction to Real Analysis" (Fourth Edition) b Robert G Bartle and Donald R Sherbert ...
I am focused on Chapter 3: Sequences and Series ...
I need help in fully understanding the proof of Theorem 3.4.11 ...
Theorem 3.4.11 and the start of its proof read as follows:https://www.physicsforums.com/attachments/7317In the above proof by Bartle and Sherbert we read the following:
" ... ... If $$\epsilon \gt 0$$, then the fact that $$x^{*}$$ is an infimum implies that there exists a $$\nu$$ such that $$x^* \le \nu \le x^* + \epsilon$$. Therefore $$x^*$$ also belongs to $$V$$, ... ... "Question 1
Could someone please demonstrate rigorously that if $$\epsilon \gt 0$$, then the fact that $$x^*$$ is an infimum implies that there exists a $$\nu$$ such that $$x^* \le \nu \le x^* + \epsilon$$ ... ... ? (... also covering the case where inf $$V$$ = the minimum of $$V$$ ...)Question 2
Can someone please explain why we can then conclude that therefore $$x^*$$ also belongs to $$V$$ ... ... ?Help will be appreciated ...
Peter
===========================================================================It may help readers of the above post to have Bartle and Sherbert's definition of limit superior and limit inferior ... which include the definition of the set V ... as follows ... ...View attachment 7318
I am focused on Chapter 3: Sequences and Series ...
I need help in fully understanding the proof of Theorem 3.4.11 ...
Theorem 3.4.11 and the start of its proof read as follows:https://www.physicsforums.com/attachments/7317In the above proof by Bartle and Sherbert we read the following:
" ... ... If $$\epsilon \gt 0$$, then the fact that $$x^{*}$$ is an infimum implies that there exists a $$\nu$$ such that $$x^* \le \nu \le x^* + \epsilon$$. Therefore $$x^*$$ also belongs to $$V$$, ... ... "Question 1
Could someone please demonstrate rigorously that if $$\epsilon \gt 0$$, then the fact that $$x^*$$ is an infimum implies that there exists a $$\nu$$ such that $$x^* \le \nu \le x^* + \epsilon$$ ... ... ? (... also covering the case where inf $$V$$ = the minimum of $$V$$ ...)Question 2
Can someone please explain why we can then conclude that therefore $$x^*$$ also belongs to $$V$$ ... ... ?Help will be appreciated ...
Peter
===========================================================================It may help readers of the above post to have Bartle and Sherbert's definition of limit superior and limit inferior ... which include the definition of the set V ... as follows ... ...View attachment 7318
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