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 1.8.15 ... ...

Duistermaat and Kolk's Theorem 1.8.15 and its proof read as follows:

In 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