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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 \(\displaystyle r = \text{min} \{ r_1, r_2, \ ... \ ... \ r_n \}\) we see that \(\displaystyle B_r (a) \subset U_j \text{ for each } j = 1,2, \ ... \ ... n\) ... ... "Although it seems plausible ... I do not see ... rigorously speaking, why the above statement is true ...

Can someone demonstrate rigorously that letting \(\displaystyle r = \text{min} \{ r_1, r_2, \ ... \ ... \ r_n \}\) ...

... implies that \(\displaystyle B_r (a) \subset U_j\) for each \(\displaystyle j = 1,2, \ ... \ ... n\) ... ... Surely it is possible that \(\displaystyle B_r (a)\) lies partly outside some \(\displaystyle U_j\) ... ...

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 \(\displaystyle r = \text{min} \{ r_1, r_2, \ ... \ ... \ r_n \}\) we see that \(\displaystyle B_r (a) \subset U_j \text{ for each } j = 1,2, \ ... \ ... n\) ... ... "Although it seems plausible ... I do not see ... rigorously speaking, why the above statement is true ...

Can someone demonstrate rigorously that letting \(\displaystyle r = \text{min} \{ r_1, r_2, \ ... \ ... \ r_n \}\) ...

... implies that \(\displaystyle B_r (a) \subset U_j\) for each \(\displaystyle j = 1,2, \ ... \ ... n\) ... ... Surely it is possible that \(\displaystyle B_r (a)\) lies partly outside some \(\displaystyle U_j\) ... ...

Help will be appreciated ...

Peter