
MHB Master
#1
February 10th, 2018,
00:18
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

February 10th, 2018 00:18
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MHB Master
#2
February 10th, 2018,
11:19
Hi Peter,
It’s almost immediate if you know what continuity of the Euclidean norm means: For any $\mathbf{c}\in \Bbb R^n$ and any sequence $(\mathbf{c}_k)\in \Bbb R^n$ converging to $\mathbf{c}$, $\\mathbf{c}_k\ \to \\mathbf{c}\$. Take $\mathbf{c}_k = f(\mathbf{x}_k)  f(\mathbf{y}_k)$, so $\mathbf{c} = 0$. Since $\\mathbf{0}\ = 0$, $\f(x_k)  f(y_k)\ \to 0$.

MHB Master
#3
February 11th, 2018,
02:31
Thread Author
Originally Posted by
Euge
Hi
Peter,
It’s almost immediate if you know what continuity of the Euclidean norm means: For any $\mathbf{c}\in \Bbb R^n$ and any sequence $(\mathbf{c}_k)\in \Bbb R^n$ converging to $\mathbf{c}$, $\\mathbf{c}_k\ \to \\mathbf{c}\$. Take $\mathbf{c}_k = f(\mathbf{x}_k)  f(\mathbf{y}_k)$, so $\mathbf{c} = 0$. Since $\\mathbf{0}\ = 0$, $\f(x_k)  f(y_k)\ \to 0$.
Thanks Euge ...
Appreciate the help ...
Peter