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Open Subsets in a Metric Space ... Stromberg, Theorem 3.6 .. ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
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:



Stromberg - Theorem 3.6 ... .png



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
 

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
The Theorem makes reference to "Definition 3.3" but you don't give us that definition. Also you are asking about "[tex]B_r(a)[/tex]" but don't tell us what that means. Since the theorem is about "open sets", I strongly suspect, but can't be sure, that "Definition 3.3" is the definition of "open set" and that "[tex]B_r(a)[/tex]" is the "open ball" centered at a with radius r.

If that is correct then [tex]B_r(a)[/tex] is [tex]\{ p| d(p, a)< r\}[/tex], the set of all points whose distance from point a (the center of the ball) is less than r (the radius of the ball). From that it follows immediately that if [tex]r_1< r_2[/tex] then [tex]B_{r_1}(a)\subset B_{r_2}(a)[/tex]- the smaller radius ball is inside the larger radius ball.
 

Olinguito

Well-known member
Apr 22, 2018
251
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 \(\displaystyle j = 1,2, \ ... \ ... n\) ... ...
$r$ is the minimum of all the $r_j$ and so $r\leqslant r_j$ for all $j$; hence $B_r(a)\subseteq B_{r_j}(a)\subset U_j$ for all $j=1,\ldots,n$.
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
$r$ is the minimum of all the $r_j$ and so $r\leqslant r_j$ for all $j$; hence $B_r(a)\subseteq B_{r_j}(a)\subset U_j$ for all $j=1,\ldots,n$.




Thanks for the help Olinguito ...

Peter

- - - Updated - - -

The Theorem makes reference to "Definition 3.3" but you don't give us that definition. Also you are asking about "[tex]B_r(a)[/tex]" but don't tell us what that means. Since the theorem is about "open sets", I strongly suspect, but can't be sure, that "Definition 3.3" is the definition of "open set" and that "[tex]B_r(a)[/tex]" is the "open ball" centered at a with radius r.

If that is correct then [tex]B_r(a)[/tex] is [tex]\{ p| d(p, a)< r\}[/tex], the set of all points whose distance from point a (the center of the ball) is less than r (the radius of the ball). From that it follows immediately that if [tex]r_1< r_2[/tex] then [tex]B_{r_1}(a)\subset B_{r_2}(a)[/tex]- the smaller radius ball is inside the larger radius ball.



Thanks for the help HallsofIvy ...

Peter