Is this assumption redundant in the proof of Taylor theorem

Question

I'm reading Taylor theorem in textbook Analysis I by Amann.

and its proof:

Here is the Theorem 2.18 used in the proof:

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I read the proof carefully, but could not find where the authors apply the assumption that $D$ is convex. So I think that this assumption is unnecessary.

Could you please confirm if my understanding is correct? Thank you so much!

Answer 1

It definitely comes into play in the definition of $h$, as you assure that $a + t(x-a) $ $\in D$