MHB Compactness and Uniform Continuity in R^n .... .... D&K Theorem 1.8.15

[Math Amateur]

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:View attachment 7754In the above proof we read the following:

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

Peter

 

[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 $\|\mathbf{0}\| = 0$, $\|f(x_k) - f(y_k)\| \to 0$.

 

[Math Amateur]

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