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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 \(\displaystyle ( x_k )_{ k \in \mathbb{N} }\) be a sequence of points in \(\displaystyle F\) with the property that \(\displaystyle ( f( x_k ) )_{ k \in \mathbb{N} }\) is convergent with limit \(\displaystyle b \in \mathbb{R}^p\). ... ... "My question is as follows:

How do we be sure that such a sequence exists in \(\displaystyle F\)?Help will be appreciated ... ...

Peter