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

I need some help with an aspect of Definition 9.1.3 ...

Definition 9.1.3 and the relevant accompanying text read as follows:

https://www.physicsforums.com/attachments/7865

View attachment 7866

At the top of the above text, in Definition 9.1.3 we read the following text:

" ... ... there exists a vector \(\displaystyle f'(a)\) in \(\displaystyle \mathbb{R}^n\) ... ... "My question is as follows:

How (arguing from the definition of derivative) do we indicate\demonstrate\prove that \(\displaystyle f'(a) \in \mathbb{R}^n\) ... ...?Hope someone can help ... ...

Peter