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**Derivative of a Vector-Valued Function of a Real Variable - Junghenn Propn 9.1.2 ...**

I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ...

I am currently focused on Chapter 9: "Differentiation on \mathbb{R}^n"

I need some help with the proof of Proposition 9.1.2 ...

Proposition 9.1.2 and the preceding relevant Definition 9.1.1 read as follows:

https://www.physicsforums.com/attachments/7861

View attachment 7862In the above text from Junghenn we read the following:

" ... ... The assertions follow directly from the inequalities

\(\displaystyle \left\vert \frac{f_j ( a + h ) - f_j (a)}{ h } - x_j \right\vert^2 \le \left\| \frac{ f( a + h ) - f(a) }{ h } - ( x_1, \ ... \ ... \ , x_m ) \right\|^2 \)

\(\displaystyle \le \sum_{ i = 1 }^m \left\vert \frac{f_j ( a + h ) - f_j (a)}{ h } - x_j \right\vert^2 \) ...

... ... "

Can someone please show why the above inequalities hold true ... and further how they lead to the proof of Proposition 9.1.2 ... ...Help will be much appreciated ...

Peter

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