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I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand the proof of Theorem 3.6 on page 94 ... ... Theorem 3.6 and its proof read as follows:
View attachment 9114
In the above proof by Stromberg we read the following:
" ... ... Letting $$r = \text{min} \{ r_1, r_2, \ ... \ ... \ r_n \}$$ we see that $$B_r (a) \subset U_j \text{ for each } j = 1,2, \ ... \ ... n$$ ... ... "Although it seems plausible ... I do not see ... rigorously speaking, why the above statement is true ...
Can someone demonstrate rigorously that letting $$r = \text{min} \{ r_1, r_2, \ ... \ ... \ r_n \}$$ ...
... implies that $$B_r (a) \subset U_j$$ for each $$j = 1,2, \ ... \ ... n$$ ... ... Surely it is possible that $$B_r (a)$$ lies partly outside some $$U_j$$ ... ...
Help will be appreciated ...
Peter
I am focused on Chapter 3: Limits and Continuity ... ...
I need help in order to fully understand the proof of Theorem 3.6 on page 94 ... ... Theorem 3.6 and its proof read as follows:
View attachment 9114
In the above proof by Stromberg we read the following:
" ... ... Letting $$r = \text{min} \{ r_1, r_2, \ ... \ ... \ r_n \}$$ we see that $$B_r (a) \subset U_j \text{ for each } j = 1,2, \ ... \ ... n$$ ... ... "Although it seems plausible ... I do not see ... rigorously speaking, why the above statement is true ...
Can someone demonstrate rigorously that letting $$r = \text{min} \{ r_1, r_2, \ ... \ ... \ r_n \}$$ ...
... implies that $$B_r (a) \subset U_j$$ for each $$j = 1,2, \ ... \ ... n$$ ... ... Surely it is possible that $$B_r (a)$$ lies partly outside some $$U_j$$ ... ...
Help will be appreciated ...
Peter