A question about the proof of mean value theorem for vector valued functions

Question

I'm reading a proof of mean value theorem for vector valued functions in textbook Analysis I by Amann.

I could not understand why $\|f(t)-f(a+\varepsilon)\| \leq\|f(t)-f(s)\|+\alpha(s-a-\varepsilon)$ for $t \in (s,b)$.

I only get that: because $t > s$, $t \notin S$. So $\|f(t)-f(a+\varepsilon)\| > \alpha(s-a-\varepsilon)$ for $t \in (s,b)$.

Could you please elaborate more on this point? Thank you so much!

Answer 1

$\|f(t)-f(a+\epsilon)\| \leq \|f(t)-f(s)\|+\|f(s)-f(a+\epsilon)\|$. Since $s \in S$ the second term does not exceed $\alpha (s-a-\epsilon)$.