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

I need help with an aspect of the proof of Theorem 1.8.15 ... ...

Duistermaat and Kolk's Theorem 1.8.15 and its proof read as follows:View attachment 7754In the above proof we read the following:

" ... ... The continuity of the Euclidean norm the gives \(\displaystyle \lim_{{k}\to{\infty}} \mid \mid f(x_k) - f(y_k) \mid \mid = 0 \) ... ... "Can someone please explain ... and also show rigorously ... how/why the continuity of the Euclidean norm the gives \(\displaystyle \lim_{{k}\to{\infty}} \mid \mid f(x_k) - f(y_k) \mid \mid = 0 \) ... ... Help will be much appreciated ...

Peter