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

Peter

Well-known member
MHB Site Helper
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:

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
The Theorem makes reference to "Definition 3.3" but you don't give us that definition. Also you are asking about "$$B_r(a)$$" 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 "$$B_r(a)$$" is the "open ball" centered at a with radius r.

If that is correct then $$B_r(a)$$ is $$\{ p| d(p, a)< r\}$$, 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 $$r_1< r_2$$ then $$B_{r_1}(a)\subset B_{r_2}(a)$$- the smaller radius ball is inside the larger radius ball.

Olinguito

Well-known member
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
$r$ 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 "$$B_r(a)$$" 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 "$$B_r(a)$$" is the "open ball" centered at a with radius r.

If that is correct then $$B_r(a)$$ is $$\{ p| d(p, a)< r\}$$, 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 $$r_1< r_2$$ then $$B_{r_1}(a)\subset B_{r_2}(a)$$- the smaller radius ball is inside the larger radius ball.

Thanks for the help HallsofIvy ...

Peter