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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 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 7731In the above proof we read the following:
" ... ... Indeed let $$( x_k )_{ k \in \mathbb{N} }$$ be a sequence of points in $$F$$ with the property that $$( f( x_k ) )_{ k \in \mathbb{N} }$$ is convergent with limit $$b \in \mathbb{R}^p$$. ... ... "My question is as follows:
How do we be sure that such a sequence exists in $$F$$?Help will be appreciated ... ...
Peter
I am focused on Chapter 1: Continuity ... ...
I need help with an 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 7731In the above proof we read the following:
" ... ... Indeed let $$( x_k )_{ k \in \mathbb{N} }$$ be a sequence of points in $$F$$ with the property that $$( f( x_k ) )_{ k \in \mathbb{N} }$$ is convergent with limit $$b \in \mathbb{R}^p$$. ... ... "My question is as follows:
How do we be sure that such a sequence exists in $$F$$?Help will be appreciated ... ...
Peter