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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 an aspect of Lemma 1.2.5 ...

Duistermaat and Kolk"s Lemma 1.2.5 reads as follows:View attachment 9014In the above proof by Duistermaat and Kolk we read the following:

" ... ... Assertion (i) follows from Definition 1.2.2 ... ..."I have tried to demonstrate a rigorous proof of Assertion (i) but have not been happy it is fully rigorous ...

Help will be appreciated ...

Peter ========================================================================================It may help readers of the above post to have access to the start of Section 1.2: Open and Closed Sets ... which includes Definition 1.2.2 referred to above ... so I am providing aces to that text ... as follows ... View attachment 9015

View attachment 9016

Hope that helps ... ...

Peter

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

I need help with an aspect of Lemma 1.2.5 ...

Duistermaat and Kolk"s Lemma 1.2.5 reads as follows:View attachment 9014In the above proof by Duistermaat and Kolk we read the following:

" ... ... Assertion (i) follows from Definition 1.2.2 ... ..."I have tried to demonstrate a rigorous proof of Assertion (i) but have not been happy it is fully rigorous ...

**Can someone please demonstrate a fully rigorous proof of Assertion (i) ...**

Help will be appreciated ...

Peter ========================================================================================It may help readers of the above post to have access to the start of Section 1.2: Open and Closed Sets ... which includes Definition 1.2.2 referred to above ... so I am providing aces to that text ... as follows ... View attachment 9015

View attachment 9016

Hope that helps ... ...

Peter