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  1. MHB Master
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    #1
    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

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    #2
    Hi Peter,

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

  3. MHB Master
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    Quote Originally Posted by Euge View Post
    Hi Peter,

    Itís 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

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