I shall just reference another question See the proof given here. (Please see the link before reading the rest; I thought posting the same thing would be counterproductive.)
In the proof, when we take $ \beta $ to be the supremum on the left of the inequality, for $ s \in (a, t) $. Then that implies $ \beta \geq \frac{\phi(t)-\phi(s)}{t-s} = \frac{\phi(s)-\phi(t)}{s-t} $ for all $ s \in (a, t) $ $(1)$. But in the next step of the proof says that $ \beta \leq \frac{\phi(s)-\phi(t)}{s-t}$ for all $ s \in (a, b)$. This is contradicting $(1)$. This is what I am not able to see. Any help is deeply appreciated.