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I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...

I am focused on Chapter 1: Continuity ... ...

I need help with another aspect of the proof of Theorem 1.8.17 ... ...

Duistermaat and Kolk's Theorem 1.8.17 and its proof (including the preceding relevant definition) read as follows:View attachment 7765

View attachment 7766

In the last line of the above proof we read the following:

" ... ... and so \(\displaystyle y \in \{ x \in \mathbb{R}^n \ \mid \ \mid \mid x - y \mid \mid \lt \frac{1}{ j_0 } \} \subset \mathbb{R}^n\text{\\} K\). ... ... "Presumably \(\displaystyle y \in \{ x \in \mathbb{R}^n \ \mid \ \mid \mid x - y \mid \mid \lt \frac{1}{ j_0 } \} \)

... because \(\displaystyle \mid \mid y - y \mid \mid = \mid \mid 0 \mid \mid \lt \frac{1}{ j_0 }\) ...

Is that right?

BUT ...

How/why does \(\displaystyle y \in \{ x \in \mathbb{R}^n \ \mid \ \mid \mid x - y \mid \mid \lt \frac{1}{ j_0 } \} \subset \mathbb{R}^n\text{\\}K\) mean that \(\displaystyle \mathbb{R}^n\text{\\} K\) is open?Help will be much appreciated ...

Peter

I am focused on Chapter 1: Continuity ... ...

I need help with another aspect of the proof of Theorem 1.8.17 ... ...

Duistermaat and Kolk's Theorem 1.8.17 and its proof (including the preceding relevant definition) read as follows:View attachment 7765

View attachment 7766

In the last line of the above proof we read the following:

" ... ... and so \(\displaystyle y \in \{ x \in \mathbb{R}^n \ \mid \ \mid \mid x - y \mid \mid \lt \frac{1}{ j_0 } \} \subset \mathbb{R}^n\text{\\} K\). ... ... "Presumably \(\displaystyle y \in \{ x \in \mathbb{R}^n \ \mid \ \mid \mid x - y \mid \mid \lt \frac{1}{ j_0 } \} \)

... because \(\displaystyle \mid \mid y - y \mid \mid = \mid \mid 0 \mid \mid \lt \frac{1}{ j_0 }\) ...

Is that right?

BUT ...

How/why does \(\displaystyle y \in \{ x \in \mathbb{R}^n \ \mid \ \mid \mid x - y \mid \mid \lt \frac{1}{ j_0 } \} \subset \mathbb{R}^n\text{\\}K\) mean that \(\displaystyle \mathbb{R}^n\text{\\} K\) is open?Help will be much appreciated ...

Peter

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