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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 1.8.6 ... ...

Duistermaat and Kolk"s Theorem 1.8.6 and the preceding definition regarding proper mappings read as follows:View attachment 7732In the above proof we read the following:

" ... ... Thus for \(\displaystyle k\) sufficiently large, we have \(\displaystyle f( x_k ) \in K = \{ y \in \mathbb{R}^p \mid \ \mid \mid y - b \mid \mid \le 1 \}\), while \(\displaystyle K \) is compact in \(\displaystyle \mathbb{R}^p\). ... ... "I am confused by the above statement ... can someone please explain/clarify ... ...

Apologies in advance if I am missing something simple ... ...

Note that in particular I do not quite understand the statement \(\displaystyle f( x_k ) \in K = \{ y \in \mathbb{R}^p \mid \ \mid \mid y - b \mid \mid \le 1 \}\) ... ... Hope someone can help ...

Peter***EDIT***

Oh ... !

\(\displaystyle f( x_k ) \in K = \{ y \in \mathbb{R}^p \mid \ \mid \mid y - b \mid \mid \le 1 \}\)

... probably means f( x_k ) \in K

Is that right?

But then ... why is \(\displaystyle K\) compact?

and

... why does \(\displaystyle x_k \in f^{-1} (K) \cap F\) ... ... and further, why is \(\displaystyle f^{-1} (K) \cap F\) compact?Hope someone can help with these further questions ...

Peter

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

I need help with another aspect of the proof of Theorem 1.8.6 ... ...

Duistermaat and Kolk"s Theorem 1.8.6 and the preceding definition regarding proper mappings read as follows:View attachment 7732In the above proof we read the following:

" ... ... Thus for \(\displaystyle k\) sufficiently large, we have \(\displaystyle f( x_k ) \in K = \{ y \in \mathbb{R}^p \mid \ \mid \mid y - b \mid \mid \le 1 \}\), while \(\displaystyle K \) is compact in \(\displaystyle \mathbb{R}^p\). ... ... "I am confused by the above statement ... can someone please explain/clarify ... ...

Apologies in advance if I am missing something simple ... ...

Note that in particular I do not quite understand the statement \(\displaystyle f( x_k ) \in K = \{ y \in \mathbb{R}^p \mid \ \mid \mid y - b \mid \mid \le 1 \}\) ... ... Hope someone can help ...

Peter***EDIT***

Oh ... !

\(\displaystyle f( x_k ) \in K = \{ y \in \mathbb{R}^p \mid \ \mid \mid y - b \mid \mid \le 1 \}\)

... probably means f( x_k ) \in K

*\(\displaystyle K = \{ y \in \mathbb{R}^p \mid \ \mid \mid y - b \mid \mid \le 1 \}\)***WHERE**Is that right?

But then ... why is \(\displaystyle K\) compact?

and

... why does \(\displaystyle x_k \in f^{-1} (K) \cap F\) ... ... and further, why is \(\displaystyle f^{-1} (K) \cap F\) compact?Hope someone can help with these further questions ...

Peter

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