I'm trying to prove Lemma 3.7 in this paper, i.e.,
Lemma Let $T \in [0, \infty)$ and $f:[0, T] \to \mathbb R$ be differentiable. If $l \in (0, \infty)$ such that $\{t \in [0, T] ; f(t) > l\} \subset \{t \in [0, T] ; f'(t) <0\}$, then $\sup_{t \in [0, T]} f(t) \le \max \{f(0), l\}$.
Could you confirm if my below attempt is fine?
Proof Assume the contrary that there is $t_0 \in [0, T]$ such that $f(t_0) > \max \{f(0), l\}$. Then $t_0 >0$ and there is $t_1 \in [0, t_0]$ such that $f(t_1) = \sup_{t \in [0, t_0]} f(t)$. Then $f(t_1) > l$. So $t_1 >0$ and $f'(t_1) <0$. This implies there is $t_2 \in [0, t_1]$ such that $f(t_2) > f(t_1)$. This is a contradiction.